So, my playing has been coming along pretty well - I'm building accuracy very quickly and I learn new things every day - but there's one thing that's really hurting my sound. Whenever I pick, say, the high E string, the low strings quietly ring out as well, even though I didn't even touch them at all. It's making my playing sound really sloppy when I don't think it is.

So, is this a problem with my guitar or do all guitars do this and it's something I'm going to have to get used to muting? Like I said, I'm sure it's not my finger catching the string or something silly like that, because I'm 100% sure that my hand is not touching it, and the problem skips to strings that aren't even adjacent to the one I picked.

Thanks in advance for your answers, and rock on!
Yeah, those other strings are ringing sympathetic notes and/or harmonics. If you hit the high E, the vibration will go through the body and ring the low E, the E harmonic on the A and so on. You just gotta learn when to mute the strings.
i know that my cheap electric guitar does just that, and my more expensive borrowed guitar doesn't. so yea it has to do with your guitar, but you should still be able to work a technique to mute those extra strings, at least most of the time.
just BARELY rest your palm over the bridge so it kinda, halfway mutes your strings. does wonders and is hardly noticeable.
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Half guitar, half technique. The guitar will vibrate and other strings will vibrate, but some guitars don't do this (as much). So learn to mute the strings slightly and you will be better no matter what guitar you use.
Quote by -BlackZeppelin-
Psh. I can can play fleshlight acoustically.

With your picking hand, try using the edge of your thumb to mute the strings above the one you are picking.

With your left hand, lay a finger lightly across the strings you are not playing, especially the ones below the string you are picking. I often do this with the index finger (like a barre chord, but with less pressure) or the pinky.

It sounds like a pain and can slow you down at first, but it really cleans up your sound -- especially if you like to play with high gain.