Beginner guitar player here. Sometimes when I try to play even simple chords (such as a C chord), I am constantly getting buzzing from either the string above or below it. I understand this happening if my finger was literally touching the other strings, but this happens even if I'm a hair's width away from it. Is this a technique issue? If so, does anyone have any advice or know of free resources for how to better angle and train the left hand to bend the way it's supposed to for eliminating the buzz of neighboring strings?
Well that might be exactly the problem - your finger is so close to the other strings that when you strum the vibrating sting is hitting the finger.
Ideally your fingers should be perpendicular to the fretboard.
Does it go away when you play the open-c chord at the 13th fret? (Literally slide up the chord so you're fretting the high-C note on the B string, 13th fret.) The fretboard is wider so you can maybe get a better feel for what's going on.
Is there a nice space gap between your palm and the neck so you have lots of room to reach?
I'm also assuming you're fretting the notes close enough to the fret. Here's a nice video on that.
Remember the strings vibrate so if you are indeed a hairs breadth away from the next string it will touch your finger as it moves. You need to place your finger so it is away from the next door strings by changing your positioning of the hand so it is the finger tips pressing down the string. It is all too easy to collapse your fingers or put them diagonally so they interfere with other strings. Make sure your thumb is in the correct position behind the neck as this rotates your hand around the neck allowing the fingers to be more perpendicular rather than collapsed. It also helps to hold the guitar at the correct angle. Beginners tend to turn the guitar to see what they are doing which means the hand has farther to stretch round.
Lots of things can cause strings buzzing on the frets. If you're not positioning your fretting fingers as close behind the frets as you can without putting your fingers on the frets, that can contribute to the problem. Move your fingers up to the frets without touching them and see if that helps eliminate the problem. You also might have some loose or high frets, fixable at your local guitar shop. But most likely the guitar needs a set-up, which your local guitar shop can also take care of for you. If it's a set-up you need and you get it done, you'll find it easier and more fun to practice, learn, and perform and the fret buzz will probably be history for you