Whoa!? Finally I see someone that reads the fuckin manual, or is smart to begin with! Cool. You did exceptionally well - downloading gimp, and figuring out about layers in one day. Most people I see elswhere on other forums don't even know where to download it after I direct them to gimp.org.
Anyhow, PS has a function that allows you to save it into a "web friendly file", which cuts down the file size. With Gimp I haven't come across such a function,
but - it is easy as to get the same result.
First off, your avatar need not be a .png make it a gif or jpeg file. PNG is a good file format for files that you want to have a maximum quality with, simply because when you use a more compressed file format, like jpeg, your image looses quality - which is not visible to the naked eye. Only when you zoom in by about 4x -- 8x you can begin to see the difference.
So... save it into .jpg format. See if that cuts down the size. If it ain't, save your file, after you've merged all layers (layers > merge layers - or something like that), save it into the native Gimp file format (for PS it's .psd), which is .xcf. You will notice that suddenly your file is either much bigger or much smaller. Nevermind what. Now open that xcf file again and save it into jpg.
Done. That should do the trick. If not, whilst exporting, just grab the quality bar and decrease it by about 15%. Won't be visible to the naked eye, but will
definitely decrease the file size immensly.
Cheers.
PS - check out GUG, in david's post above. *Extremely* handy for any Gimp users.
Ohh, yeah -- welcome into graphics (and Open Source software)!

Hope you enjoy it. =]