(2-part post):
As VHG suggests, Photoshop (which,of course, is from Adobe) is king of the photo-editing programs. However I wouldn't recommend using it unless you're prepared to buy at least one really good book to teach you the basics. I've used Photoshop for several years (and I've got several books about it) but I've still got a lot to learn! (I use Photoshop 6. Photoshop CS is probably the nearest current equivalent but it's certainly not cheap. Photoshop Elements is more basic but will fulfil most people's needs).
The nearest free alternative to Photoshop is GIMP. Once again, it takes quite a while to get used to using it (and I'd regard it as essential to download the documentation as well as the program) but it might be worth considering:
http://www.gimp.org/
VHG has already provided a link to Serif's website, where you can download the excellent PhotoPlus SE for free. Alternatively, you can get the full 'bells and whistles' version (PhotoPlus 9 for just �9.99).
It's harder to recommend web design software since there are basically three different approaches. Firstly, you can learn HTML coding and use any text editor (such as Notepad) to produce your pages. 'Purists' often recommend that technique but others prefer to use software that handles all of the HTML coding for you. Such software is divided into programs where you start from a blank page and others where you're provided with templates to adapt to your own purposes. (Obviously some programs will let you do both).