I've tried all different types of picks and sizes, as well as different ways of holding it. Nothing seems to work.

I'm a beginner at guitar, and alternate picking, and am trying to learn out of Troy Stetina's book "Speed Mechanics for Lead Guitar". He stresses the importance of alternate picking, especially between strings.

I'm a beginner, and I suck, but the picking in general just doesn't feel right and I always miss notes. How should I hold the pick?

Currently I can do with just thumb and index finger, which feels easier but is more sloppy than using my thumb, index, and middle finger. My three finger combo gets in the way sometimes when I'm going up and downstrings (since my middle finger is being used, it'll hit extra strings.)

what can I do so I stop becomming so frustrated and can learn how to pick right?
slow down. hold the pick whatever way is most comfortable it really doesn't matter much. it would be a little stupid holding it with your index finger and your pinky you know? but other than that it shouldn't matter much.

you just need to slow down the exercises you're doing (use a metronome) and gradually build speed

there is a speed where you can pick everything perfectly, start from that
If you're truly a beginner you should try to learn songs with alternate picking.
Thats the way I learnt it. And its much more fun since you get to learn a great song and get better at playing. And remember, challenge yourself!
dude, hold it however you want to hold it. its not the way you're holding it thats making you mess up. you say you're a beginner. juss start out slowly, as slow as you need to, then gradually speed up. it WILL take time. people always want instant results, and that juss doesnt happen with guitar. you wont even notice when you can suddenly do it.
What songs do you recommend? I'm into metal and rock. Most songs I try to play are from Megadeth which are too hard.

For the picking thing, how do you strike the strings? Do you kind of scrape it with the pick turned, or hit it parallel?
A great song for that is Radiohead - Street Spirit. Practise it really slowly accurately, then play with dotted rhythms, and work up to a fast speed
Don't overlook the fact that if it feels awkward and unnatural to you that you can use economy picking. I never really liked the feel of alternate picking but i can pick just as fast if not faster when changing between strings with economy picking.
Art is making something out of nothing and selling it - Frank Zappa
I just wanted to alternate pick to get better. Well, playing guitar better anyway. I just feel like some positions feel better than others.

One more question:

how close to the bottom of the pick do you hold it? How much is showing, and at what angle do you pick against the strings?

Personally if your that uncomfotable with picking I would practice downstrokes on the low E string along with a metronome before I tackled alternate picking. For a great alternate picking riff try the intro to Crazy train. Generally most players hold the pick between thumb and index finger. There are lots of other issues with picking included how to mute, whether to anchor your fingers on the body or rest your palm on the bridge etc. It takes some experimentation and practice to determine what works best for you - lots of great players have different methods - if it sounds good and works for you use whatever feels right.

Generally when you are picking single strings that require any amount of speed the goal is to strike the string more or less horizontally ( although people vary this too) with the tip of the pick. Try using just the vary tip. Keep practing until you produce a good clean sound.
Good luck and be patient.
With alternate picking, do you alternate based on rhythm, or how close the strings are?

For example, on that Street Spirit song, the last three notes seem hard to maintain with rhythm. String 4 is a downstroke, while string 5 is an upstroke, and then string 4 is a downstroke. Should I vary string 4 to an upstroke so I can hit 5 with a downstroke, or does it really matter once I get going?