Web sites will have pictures on. Backgrounds, photographs and animations. This site is about pages not graphics but it might help to look at some of these applications so you can do the basics.
Adobe Photoshop is the obvious Windows package for graphics. It will do most things you want it to do so if you want to buy it (or already have it) stick with it. There are free alternatives though.
Resizing and tweaking images
This is a common task and an easy one. Any graphics package will probably do this but Irfanview will allow you to simply select and crop or to enhance colours, brightness and contrast. It will also load most image formats and save them as whatever format you want. Irfanview is mainly a viewing program (you can watch videos or listen to MP3s on it) but also has some editing ability.
Creating images
The Gimp is a free alternative to Photoshop. It can do almost anything that Photoshop can do but it tends to be a lot harder to work out how. If you are willing to search the Web for HowTo guides then you will be able to do almost anything with Gimp. Another advantage is that it works on Linux and Mac as well as Windows. It will allow you to create drawings, edit and enhance existing images. It can crop images and understands almost any graphical format there is. It also has built in processing which can be used to create buttons, tiled backgrounds etc..
Creating animated GIFs
Gimp will do this and so will Photoshop. Another alternative is Microsoft GIFAnimator.
Choosing good colours
There are a number of sites and tools which offer you suggestions about how well certain colours go together. Just search for colour wheels or complimentary colours. Other sites, such as this one suggest colours which will work well with a given image.
