To get started, watch this video.
For inspiration on some examples of logos, check out this page.
Go ahead and get started on your logo design (512px by 512px)! Your first step might be to pick a small set of colors to work with. Remember that it needs to be 100% your own work and don’t get discouraged if I come back and ask you to keep working and improving on your design. Start small, keep it simple, and have fun!
Of you’re having trouble figuring out Adobe Illustrator, Adobe TV has a bunch of videos that may be of help.
Below are some how-to videos for some Illustrator basics.
Create a new file
Set a background color with the rectangle tool