If one is serious about digital photography, Adobe Photoshop CS4 is worth every penny. Students can purchase it at a discounted price on JourneyEd.com or adobe.com. Adobe Photoshop CS5 is coming out in few weeks and Amazon is taking pre-orders.
If one is only semi-serious, Adobe Photoshop Elements 8 is a great piece of software...which is a poor man's version of Photoshop. Learning to use Photoshop Elements will help transition to Photoshop later. PSE 8 can be had for about $50 - $60 (after mail-in-rebate). Checkout amazon.com and other online retailers...or even eBay. While it is inexpensive, PSE 8 is still very powerful and it may be all the software a photographer ever needs! I am by no means a photographer but I am still using Adobe Photoshop Elements 6 which I bought over two years ago when I do not have access to home PC which has Adobe Photoshop CS4 but have my old laptop with me.
If one is only interested in adjustments to an image (after the image is captured in camera), Adobe Lightroom is amazing. Adobe Lightroom 2 is the current version. Adobe Lightroom 3 is coming out in few weeks. It is currently in beta and can be downloaded and played with for couple months.
All the above applications allow cataloging (organizing/tagging/searching of images) on a computer.
Picasa (or some such free software) may be good for really basic image adjustment/editing, but PSE or Lightroom or Photoshop is what a student of digital photography ought to consider.
I've used gimp on Linux in the past. It is now available on Windows XP as well. This should be considered a better alternative to Picasa if one's needs go beyond the basic editing. Gimp is also free!
Lastly...some (any) image editing (using any software, even Picasa) before posting/emailing/printing the images is better than no image editing at all.
There are several books on Photoshop and Photoshop Elements. I like these:
The Adobe Photoshop CS4 Book for Digital Photographers by Scott Kelby
The Photoshop Elements 8 Book for Digital Photographers by Scott Kelby
The Adobe Photoshop Lightroom 2 Book for Digital Photographers by Scott Kelby
There is also a set of books of common-sense tips on digital photography by Scott Kelby: Digital Photography Boxed Set, Volumes 1, 2, and 3. Even if you're only a digital photography enthusiast, you should give this set a look. Your local public library may have it.