Swells
A Swell or volume swell is a technique used to gradually increase volume, masking the transients associated with the attack portion of a sounded note.
Volume Pedal or Knob
The simplest way to do this is by using a volume pedal. You simply move the pedal fully to the heel position (no volume) before sounding the note. Then pick or otherwise sound then note and quickly begin pushing the pedal to the toe position (full volume).
You can do it identically with your guitar's volume knob. Depending on how you hold the pick, the location of the volume knob, and your ability to manipulate the volume knob while you pick or quickly move from picking to working the volume knob, it can prove a bit daunting. At first it will feel ridiculously difficult. Like any technique, you must start slow and build up your ability. Make sure your knob's rotation is nice and smooth and you can start turning it quickly after picking.
How quickly you go all the way from no volume to full volume is up to you. But if you go too quickly right after sounding the note, you may get some of the transient. You can diminish this by using cleaner picking technique, finger-picking, or higher guitar action. Or you can begin your swell later, or slow the speed of the swell.
Feedback
If you are playing with your monitors or amp and cab cranked up loud enough and/or using enough compression (which is a natural part of most distortions), you may get feedback. It will depend on your location in the room, the position and angle of how the guitar is held in relation to the speakers. Naturally, the closer you get, the more feedback you get.
But all feedback is not equal. Often you get harmonics rather than fundamentals. And it's often impossible to determine which harmonics you will get. This makes intervals and chords almost always a disaster. Feedback has to be controlled. All the open strings must be muted well, so only the desired note or one of its harmonics rings out. Using too much gain or being too close to the speakers will make even muted strings feedback and squeal. You can roll back the volume knob or step back to find the sweet spot.
Rather than sustaining into feedback, you must induce the feedback to get a swell. Sometimes it's easy enough to just fret the note and mute the open strings, then move closer to the speakers. But sometimes you need to give it a nudge. A slight hammer-on often works. Or applying slight vibrato.
Specialty Gear
The Sustainiac is a pickup system designed to induce feedback. Unlike the chaos of a speaker/string-resonance feedback, it is much more controlled and musical. It has two modes - one is designed to simply sustain or induce whatever note you're playing. The other adds harmonic overtones, sounding more similar to speaker feedback. Either way you get a nice swell, but you still can't use it for chords or intervals.
The E-Bow is a cool device that's supposed to emulate the sound of using a bow on a guitar, similar to violin. It avoids a sharp, transient-laden attack you get from a playing with a pick. The downside is still that you can only use it on a single string at a time, and it has a bit of volume to it, making it impractical to use in parallel or interspersed with picking.