burrs.....

I think everyone has that same problem when they first start out, although it may be more difficult for some.  The only advice I can give is to stretch out your fingers and to try as hard as possible to use the very tips of your fingers (for open chords, which I'm assuming is what you're learning, ie G, A, C, D, E, and their variants).  For me, I kept practicing that, and eventually (I didn't really notice) my chords just all of a sudden were ringing clear.  Good luck!

Hello Sinann, I'm probably at the same stage as you having recently picked a guitar up after 20 or so years (I wasn't any good back then either)!

I also have the ocasional buzz and also have a cheap guitar.  I found Justinguitar.com great to learn and he has a good technique when practising chords, simply strum the chord and then pick each individual string and this helps me identify which string was buzzing.  In my case the culprit was a different string on different chords so adjusting the fingers was easier.

I'm getting similar problems with the 5th and 6th strings since re-stringing my guitar. Been playing for about a year without any buzzing prior to this. I'm guessing this is due to increased tension in the new strings and that I just need to work on my strength and technique.