The buzzing when you strum could be coming from playing too close to the frets or not putting enough pressure on the fretboard with your left hand. If you do not have calisouses on your fingers yet, it may take a little while to develope strength in your fingers so that you can press hard enough to get a good sound. (Once you have calisouses you shouldn't have to press hard at all)

Practice makes permanent.

You probably have always heard to practice slowly on whatever you need to work on, and this is good advice.

The previous poster had a good list of things to pay attention to. Make sure to look for a good teacher. They can give you answers on all the little things here and there that you are unsure about, as well as a good direction to take.