So, I'm pretty much completely new at designing avatars and signatures, however curiosity has gotten the best of me and now I wish to partake in graphics design as well.

Are there any websites/tutorials that would guide me in learning how to do signatures/avatars? Also, which image editor that is free would be most useful while doing this?

you can try gimp or photoshop

You can get a free trial of adobe photoshop : www.adobe.com
You can get gimp: www.gimp.org

Tutorials

http://www.gfxvoid.com/forums/forumdisplay.php?f=7- Graphic forums with a bunch of tutorials
http://good-tutorials.com/- Website with a collection of Photoshop tutorials
http://www.pixel2life.com/- Website and forum filled with basic Photoshop tutorials.
http://the-dark-arts.net/forums/index.php?showforum=6- Forum with tutorials for Photoshop and Paint Shop Pro.
http://browse.deviantart.com/resources/tutorials/digiart/?order=9&startts=1202371200&endts=1202457600- deviantART's collection of tutorials.
http://www.learnit2.com/- Website filled with Photoshop and Illustrator tutorials.
http://www.gimptalk.com/forum/- Huge forum community with tons of GIMP tutorials.
http://www.thegnomonworkshop.com/tutorials.html- Link to free tuts.
http://movielibrary.lynda.com/html/modPage.asp?ID=327- Learn how to use different programs