Artists usually create their logo artwork in an illustration program like Adobe Illustrator, Macromedia/Adobe Freehand, Corel Draw or a number of other vector based programs. (Illustrator is the most popular.) These programs have various tools that allow you to easily create all sorts of shapes. Mastering these programs to create logos similar to those you see everyday is very tough so don't expect great results immediately. By the way, sometimes artists do start their logos by drawing them on paper and then scanning them in. But after they scan them in, they basically trace the image using the tools in the illustration program. (You can't just scan it in and use it if you want it to look good.)
Once the final artwork is created in the illustration program, the artist takes into a bitmap image program like Photoshop or ImageReady to optimize it for the Web (make the file size smaller and acceptable for use over possibly slow bandwith). Logos and linedrawing artwork should be be optimized as a GIF image 99% of the time. Photos should be usually be saved as jpg images.
For more info on creating web graphics, I recommend books by Lynda Weinmann. She has a site called lynda.com that is all about web graphics. Visit adobe.com as well. They have tutorials about using their programs for designing and optimizing graphics.
In my opinion, you have to have a pretty good background in art to really be good at graphic design. Just knowing your way around programs like illustrator and photoshop is no different than not knowing English and using MS Word.
Also, it would be in your best interest to stick with the Adobe tools if you have the money. They are the industry standard every professional worth their salt uses.