If you have Photoshop available to you, it has a very intuitive vector drawing tool called the pen tool that is much easier to use than the one in illustrator. I would suggest tracing your logo at the current available resolution, and then increasing the size of the image to however big you want it. Raster images are fine, a lot of companies just require that they be on a 20ft canvas, but that is hard to do on a lot of consumer PC's. So as Madison said, try to stick with vectors; and work with small vectors at first.
If you are looking for logo inspiration or photoshop vector tutorials, Worth1000.com is a wonderful resource.
Illustratorworld.com is basically it's alternative for Adobe Illustrator.
Or if you want to stick to the free GIMP software, A Vector Sig tutorial can be found here:
http://forums.nintendo.com/nintendo/...sage.id=594359
and adapted for your needs.
And lastly, if you would like to use the Inkscape, a very easy to use set of tutorials can be found at this website:
http://k12edcom.org/?q=node/17