Seriously. Most people say between the thumb and index coming out the side of your thumb, but I keep messing up, hitting wrong strings, bad rhythm/timing, etc. Any other ways of holding a pick or angling it I should try or do I just need practice? If "just practice", please suggest any practice techniques for improving picking speed and accuracy. Some songs or licks that make use of those techniques would be especially helpful, as opposed to dry exercises.
There is no one way. I probably use 5 or 6 different ways, depending on what/where/how I'm playing.
I agree with the good Doktor - hold it tight enough so you don't drop it, loose enough to have flexibility.It also depends on whether you're stumming or picking single/double strings, skipping strings and a myriad of other ways to play.
It sounds like you haven't been playing long. Develop your own techniques, use others' ideas, but adpt them to your own playing. Try out different pick materials and designs.Also consider where on the strings you're going to pick - close to bridge, over the neck pup, and so on. Someone will no doubt say 'this is how the pick should be held' - but most likely they mean 'this is how I hold my pick.' Maybe not much help to anyone else..?
Without seeing a picture, it's hard to say exactly what you are doing, but honestly, if you look at 20 guitar greats you will find 20 different ways to hold a pick. It really doesn't matter as much as you think. You just need to make sure you can hold it tight enough to not drop it, and loose enough to be able to move it/angle it as needed. Some folks have their pick parallel to the strings, some have it almost 45 degrees across. Both have pros and cons, but learning to vary this angle becomes essential as you improve, both for speed and to avoid going deep when you are sweep picking while skipping strings etc.
I hold my pick like this for general picking, but I move it around a lot.
Yes, it is absolutely just practice. Alternate picking practice exercises tend to begin with up and down just on one string; then walking with 4 on one string, 4 on the next etc all the way across the fretboard and back; then the same but playing 2 different frets on each string, eg low E fret 5 down up, then fret 8 down up, then up to the next string and 2 on the 5th fret, then 2 on the 7th etc.
Once you can do those consistently, triplets are a good next step, playing eg down-up-down on the bottom string, before moving to the next. This gives the opportunity to understand the beginnings of sweep picking, as your initial pick on the next string can be a continuation of the previous one.
There are many YouTube based guitar teachers - have a good browse.
I know you said no dry exercises but IMHO there are some "dry" exercises you should be doing away from the guitar that will greatly help you with pick control and speed.
Back in the 1990's I was facing a crisis, of sorts. I was in a rock band and having trouble keeping up and as the lead guitarist I needed to be able to shred at least a little bit. I happened across an ad in a guitar magazine that changed my fortunes greatly. It was called the Ruebottom (sp?) Method and they sent me a VHS tape with this guy doing these simple finger exercises. I watched the video exactly once, about a five minute watch. These exercises are really that simple.
Within a few weeks I was shredding and wailing. Of course I was maybe a bit more advanced than it sounds like you may be but the point stands - this will level up your playing like nothing else if you do the exercises. IMHO right hand/pick control is the most important aspect of guitar playing, especially for beginners...and that is often overlooked.
The exercises - I'll try to explain them but I can post vids if you are interested.
Exercise 1: Start with your hand out in front of you, palm down, fingers together and thumb held out at a right angle to your hand. Like you were doing the "L" for loser sign but all of your fingers need to be side-by-side. Rotate your thumb forwards, then backwards. Count off while doing it and try to do them rhythmically in time to music you are listening to. Set your own limits on how many reps but do them every chance you get.
Exercise 2: Put your thumb and forefinger together similar to how you might hold a guitar pick but your thumb should be perpendicular to your forefinger and intersecting at the back of the knuckle closet to the fingertip. Your other fingers are curled in and touching the palm - not a fist, just clasped lightly. Rest your hand on your thigh (or some other surface in front of you) and draw your forefinger in. Notice the thumb slightly bends at this point. This is the starting position. Flick your forefinger out making sure to keep the thumb in contact with it (both thumb and forefinger move). Your fingertip/nail should slide along the surface to give a slight resistance. That's a downstroke. Flick it back to the start position remembering to make slight contact with your fingertip to the surface. That's an upstroke. You can practice them separately by not making contact with your fingertip on the off stroke or together to simulate alternate picking.
One thing about these exercises is they are simple and can be done almost anywhere. I used to do them during my daily commute - still do sometimes. These are all about muscle memory and training your fingers to move in the correct manner to pick a guitar. That movement just isn't natural to some people but these exercises can make it second nature for anyone.
Here's a short video Guitar pick hand exercises
There are a surprising number of things you can do with a pick. I recommend YouTube videos about it-- they are quite fascinating.
One common suggestion I've seen is that the pick is held such that it's not parallel to the string-- it's on an angle, and this makes it easier to slide the pick through the note, instead of it getting "caught" on the string.
Besides the other great answers, I just want to add that I once was teaching a beginner who was similarly frustrated with the pick. It turned out they were using an extra stiff pick. When they switched to a softer one they had a much easier time. I see Tim also suggested playing around with materials.