The answer depends very much on how deeply you want to go in the world of graphic design. Since you're already familiar with Photoshop and Illustrator, you should consider one of the Adobe Creative Suite bundles. If you own a Photoshop license already, you can get a discount on an upgrade to Web Standard or Web Premium.
There are two major advantages to this route: 1. The programs work extremely well together and make for a very tight workflow that gets you from start to finish more quickly than any other products out there; and 2. the web and your local bookstore are packed with learning resources at every level from beginner to advanced. There is a huge community of users to interact with, extensive tutorials on Adobe TV and from third party sites like Lynda.com, for example, which has a wide range of video tutorial courses on every program in the entire Creative Suite.
A third plus is that the most recent versions (Creative Suite 5.5) have been upgraded in key areas to make web development faster and much simpler.
For Logo work, Illustrator reigns supreme, but InDesign has many of the same capabilities as both Illustrator and Photoshop (code from Photoshop is actually baked right into InDesign), but is the king of layout. Since CS5 I have moved quite a lot of work from Photoshop to InDesign because it's much faster and more precise for many things that used to be the exclusive province of Photoshop.
If you're on a very tight budget and not considering graphic design as a profession, consider Xara (www.xara.com). The feature set is smaller, but the programs are light and fast and may do everything you need.