Choosing a Pick
There are numerous considerations for choosing a pick: shape, size, thickness, tip, material/durability, and grip. There are also special picks, such as thumb and finger picks. My advice is to buy one each of a wide variety of picks. It might cost you $20-30 for this initial order, from which you may only find 2-3 that you like. But that's better than using a suboptimal pick. The ideal pick for you is based on your personal traits like how you grip the pick, how large your hand is, how hard you grip the pick, etc., as well as related to your gear, such as string gauge and tuning, or your playing style (mostly strumming vs. lead or riff playing). Picks will also impact tone, so that has a bearing on the one(s) you settle on.
I play Dunlop John Petrucci Jazz III picks, and highly recommend them to others.
Contents
Brands
Jim Dunlop is the biggest name I know of in for picks. Fender and other guitar manufacturers make picks, as well as no-name companies whose picks are simply rebranded. Fender picks aren't bad (they are certainly pretty!), but I'd stay away from anything else. Picks are relatively inexpensive and are used as a promotional tool rather than a serious product. This is why I like Dunlop and other brands below dedicated to selling picks - they survive by making quality picks, not trying to rope you into buying something else. That said, new companies can always appear; and there are several boutique vendors that use stones and gems for ultra-durable, high-price picks. Companies like Winspear have recently emerged selling extra-thick (and expensive) picks, boasting better tone and ease of picking through each string.
Aesthetics
My advice is to try to ignore pick aesthetics entirely. You want the sound of yourself playing guitar to be beautiful, which means using a pick with excellent comfort and control. The look of a small piece of plastic that spends 99% of its time covered by your thumb/fingers is relatively insignificant.
Shape
The standard pick shape is similar to an isoseles triangle with rounded edges and even more rounded tips. The pointier of the 3 tips is what is used to pluck the strings. Most picks follow this paradigm.
Here is the shape catalog for Red Bear Trading. The first 2 picks on the third row are the "Classic" shape, as described as above. Notice the wide variety of shapes available. They can be wider or thinner and have more rounded or completely straight edges.
Shape plays an impact both on comfort and feel. By feel, I mean what it feels like moving the pick through the string during a typical picking motion. The sharper the angle of the side of the pick in relation to the string, the quicker and easier the pick will move through the string. There's a balance here: imagine playing a string with a credit card; as long as you keep the card perpendicular to the guitar body, you can't pick the string. You are just pushing on it, but you can't push through because it won't slide off the edge. But you can control the angle of the card; naturally, you will keep your hand tension roughly constant. As you push on the string, it is putting greater and greater tension on the pick (or credit card). Thus, you allow the pick to angle relative to the string, and slide off.
Picks that are narrower and hit the string with a more perpendicular edge will "catch" the string better, allowing you to really dig in and get a strong stroke. But they may require more effort per stroke and be difficult to use for fast picking. Picks that are a bit wider or have a special shape where the edges are sharply angled will slide off the string before being able to displace it much at all, leaving a weak stroke. These may allow you to quickly and easily complete strokes, but if the strokes are too weak, you won't have enough clarity in your playing.
Also pay attention to the other two tips of the triangle. Some picks are very rounded on the part opposite the tip used to pick the strings, while others are literally perfect triangles. I like to have a bit of a rounded part on top. This way, when I grip the pick, I grip on top of those two other points, but I'm not on the edge of the pick. Those two point are where you will get the most torque on the pick. Without the round extension sticking out between those points, you'd be on the edge of the pick, and that means you don't have as much torque on the other axis which keeps your pick perpendicular to the body so you can really dig into the strings when you want.
With a larger pick, you won't likely grip on the other two points of the triangle. It might not be a bit deal, but I find it a little awkward. But as mentioned, go buy a bunch of different shapes, sizes, etc. and try them out first. What's uncomfortable for some is perfect for others, which is why there is such a large variety.
Size
Picks range from smaller than a thumbprint to almost as large as your palm. Size plays into many of the things mentioned above - if a pick is large, you're going to grip it differently and that will affect how you control it.
For a given pick shape and thickness, there is a size that will work best for you, where you gain the most comfort and control. If too large, it will be awkward to grip. If too small, it will be awkward to control and maintain a grip.
Thickness
I like thick picks, and I cannot lie. Most of the picks you'll randomly encounter are pretty thin (.7 mm or smaller). In contrast, the JP Jazz III I like is 1.5 mm - over 2x as thick.
Thickness affects how much you can dig into the string during your picking stroke. Remember from Basic Technique that from a technique-perspective, you can use pick angle, the location on the pick you hit the string with, and grip strength to vary this. Well, pick thickness will override all these aspects to some degree. As mentioned above, the more pressure you apply to the string through the pick, the more pressure the string applies back to your pick and to your hand. With a thick pick, you can tighten up your grip and push really hard against that string without it sliding off the pick. When it eventually does, you'll get a big "Biiinnnngg!". With a thin pick, it doesn't matter how tight your grip is - the pick itself will bend and the string will slide off of it before you can put too much pressure on it.
So really, thick picks give you more control, but at the same time demand a bit better technique to get consistent dynamics. Thinner picks are often a bit easier for beginners, and also make strumming chords a bit easier.
Thinner picks tend to be larger and factor into account that the bend of the pick will be part of the stroke. So you generally use more of the pick tip when you pick the strings, and this means a larger picking motion per note. Thus, thinner picks tend to be worse for speed playing, requiring larger and more dedicated strokes for each note, while smaller, thicker picks can get similar pop from the string with a smaller and faster stroke.
Tip
The tip is the part of the pick you use to pluck the string. Generally, the only consideration is how sharp the tip is.
A sharp tip gives a slightly brighter (more chimey/metallic) tone, but the release as the string slides off the pick can feel a bit awkward. A very sharp tip gives you a bit more control - all the factors about your picking stroke have more impact. While a duller point is going to let the string slide off the tip once you've applied a certain amount of tension, a sharper tip lets you carry that a little further. Or you can loosen your grip to allow it to slide off.
I don't like a very sharp tip. I find it's more difficult to be consistent with timing and adds unnecessary requirements on picking technique for no real gain. I do like the tip to be a clear point, though. About as sharp as a ballpoint pen or plastic fork tip is where I like it. Also, unless you have a pick made from special material, that sharp tip is going to wear down fast, and you're left with a more standard tip, anyway.
A dull tip simply doesn't have enough control for me - I always feel like I'm sliding through each stroke, rather than being able to apply some real pop to accented notes.
Rather than go out and buy picks with all kinds of different tips, you can use a nail file to sharpen or dull a tip.
Material/Durability
Picks are made from a variety of materials. Most are plastic, but there are variants of the actual plastic used which contribute to their feel and durability. See Wikipedia - Guitar Picks for more info. Pick material will affect tone, feel, and durability.
Durability matters because those edges and tip shape you like will change as you use the pick. Eventually the pick edges may develop notches, which you have to file out to get a smooth pick stroke especially on the thinner strings. The tip may lose its point and turn into a big nub. Then your tone can get a bit dull.
For most occasions, this isn't such a big deal. Just throw the pick away and start on a fresh one. But depending on your playing style and how you set up your instrument, this may occur way too often. Plus it involves more overhead - you'd have to make sure you always have a cache of picks ready for a practice session or gig. Also, economics can come into play - you don't want to be burning through picks if they're expensive.
Metal
Some picks are made of metal. These are rigid - they have no "give" to them, similar to a rather thick plastic pick. They have a much brighter tone to them than a plastic pick. Try using a coin to pick the strings to get an idea of the tone.
The main thing here is the tone, but other considerations are also how they affect the strings causing them to wear more easily and concerns about scratching your guitar's finish.
Nylon
Nylon is a popular choice for picks - it is a very smooth material, allowing a very natural picking stroke, where the string gracefully slides off the pick edge and tip. The problem I've found is that they wear very fast. And in the case of thinner picks, while initially flexible eventually they can actually crack and break.
Tortex/Delrin
Tortex is a material made by Dunlop designed to replicate tortoise shell (see below). It is actually (I believe) Delrin, a particular kind of plastic. It promotes low friction and high durability.
I have tried quite a few Tortex picks and never really liked them particularly. They felt a little chaulky and felt like they slipped in my hand a little.
Ultex/Ultem
This is what the JP Jazz III picks are made from. It is a very hard plastic, but they seem to be easier to mold than the Tortex/Delrin ones. I've noticed Ultex picks have a wider variety of surfaces (for grip) and can be rounded across the whole body, whereas the Tortex are generally completely flat with no surfacing.
It is not indestructible - I find they wear faster than I'd like to be honest. However, they definitely last longer than the nylon ones, have a good grip, come in the size/shape I like, and are relatively cheap.
Acrylic
I believe V-Picks is the biggest manufacturer of these kind of picks. I haven't tried them myself, but they claim to be smoother and more durable than the types above. Plus they can be made clear or other unique designs. They aren't terribly expensive compared to the options below, but do cost more than your usual plastic picks. Definitely worth checking out.
Tortoise Shell
Tortoise Shell was a popular material for picks several decades ago, but harvesting and trading tortoise shell is now illegal due to wildlife preservation concners (similar to ivory in elephants and rhinos).
Red Bear actually produces picks that they claim is the same material as tortoise shell, but without harming any animals in the process. I haven't tried one of these - they are a bit price prohibitive, being around $30 per pick. Many players swear by them. I can only assume they are similar to the gemstone option below - fairly indestructable and very smooth.
The big risk here is that you're spending so much money on something so small and easily lost. You can always keep some cheap picks on hand as a backup, but because they're going to feel different, it might not be a truly comparable backup (although this is usually the case for backups).
Gemstone
I have an original Star Pick, which is made of some rare-Earth gemstone. It is ultra smooth and durable. The problem is that I only have one and am not sure if it could ever be replaced. If you are able to find one of these, I definitely recommend them. Just be sure to check reviews and make sure you are getting something actually made of real gemstone and that the durability and smoothness is up to snuff. They will not be cheap.
Again, like above, the main fear is keeping track of something small and expensive, as well as finding suitable backups.
Grip
When it comes to pick and grip, there is the right hand grip of the pick and the surface of the pick itself. This section discusses the latter.
Both material and design go into the grip of a pick. Some materials are simply slicker or less absorbant than others. This can make the pick more slick and difficult to grip, especially in humid climates or when sweating such as during a gig or intense practice session. I found nylon to be the worst material here. Tortex is likely a bit more absorbant but I still found them a bit awkward to grip.
Picks are often designed to improve grip. Methods include molding them with contours for your thumb, creating a rough or similar surface, drilling holes through them, and coating the non-tip parts in rubber. I find a subtle rough surfacing does wonders. Thumb/finger contours I never liked - they force you to grip the pick a specific way and make the picks too bulky. I've never tried holes, but I imagine they are better than nothing, although they may aggravate your skin after a while. Rubber seems like a good solution, but will also affect your right hand grip...and can add more expense per pick.
Again, buy many different options and experiment to figure out what's best for you.
Special Picks
In addition to "normal" flat picks with a single tip, there are special picks that are held or played differently than a standard picking technique.
Thumb
Some picks feature a loop of plastic you slide your thumb inside, so that you don't need to grip the pick with the index or other right hand finger. This leaves the player with the pick tone, but frees up the other fingers to finger pick strings or tap. Some players prefer it for simple comfort reasons.
I'm not a huge fan of thumb picks for a few reasons. First, you have less variety than you do with standard flat picks. Second, I don't find it comfortable to "wear" the pick. And third, I find I have a bit less control. If I grip the thumb pick like a normal pick, the last two concerns are mostly alleviated; but then the thumb pick offers no real benefit over a flat pick.
Finger
Finger picks are worn over the tip of each finger. They are used in a finger-picking technique, but give the player a picked tone rather than the duller sound of a finger.
Multi-Tip
Some picks feature multiple tips, so that each stroke actually picks the string multiple times.