I haven't noticed this till not long ago but there are a few notes that if I play on the thiner strings, the notes seem to ring out on the thicker strings. Especially fret 15 on the high e string, that one is so bad that it doesn't even sound like I've taken my finger off when I do. So I was wondering if this happens to everyone and is there anything I can do to stop it.
palm mute...maybe? I dunno, I think that should help
All you need in this life is ignorance and confidence, and then success is sure.
-Mark Twain
It's called resonance. The idea is that when one object vibrates, any close-by object "wants" to vibrate at the same frequency.

All guitars do it. You can't really do anything but pray and mute strings. Personally, I like it, since I play a lot of rhythm.

Anyways, yeah.

I had found this happened on my guitar a couple of weeks ago also. I thought that there was something wrong with my guitar. I asked a few people about it and they said it sometimes happens and I would have to mute the string(s) it happens to but me being such a beginner I don't think that I will be able to do that and still try to play the riff I am trying to play. I have only spent probably 4 hours total on my guitar. Does it just become second nature to be able to mute certain strings while playing your riff?
Well, if the riff's loud enough, you won't be able to hear the resonating strings.

What gets me is when I play on the high e, I get a resonation from the A string, while all the others are silent.

Yeah that's one of them that happens to me. The other one is when I play the B string and my low E resonates. I mean how can I mute that and the A string when I am trying to learn SCOM intro? I can barely even play the right strings at a SLOW pace.
This is one skill that all guitarists have to master, unless you plan to play like Dragonforce when they play in gigs.

It's called muting, and it only gets harder and harder as the volume and gain increases, like in a real gig.
So, practice on!