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
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 (.5 mm or smaller). In contrast, the JP Jazz III I like is 1.5 mm - over 3x 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.