Before you start work inside your computer, eliminate the hazards.
The first performance upgrade you should consider is more memory. It can double or triple the speed of bargain computers that only have 512 MB of memory. You should have at least 1024 MB for Windows XP, and 2048 MB for Vista or Windows 7. Read more about selecting and buying memory at the "Memory" page.
Upgrading your CPU is more ambitious than it used to be. It was easy to upgrade the CPU in computers built in the last century, but not the ones sold today. A worthwhile upgrade today would be expensive and difficult to do. It's usually better and cheaper to buy a new computer than to go down the CPU upgrade route.
A bigger hard drive (disk) won't make your computer run perceptively faster. That's just an urban myth. If you simply need more room, consider adding a second hard drive.
Adding a second hard drive is often easier than installing a bigger primary drive. It adds wonderful flexibility, adds capacity and redundancy for backups, and can even speed up your computer. You'd have to transfer Windows and all your files if you just get a bigger primary drive, and that's not a simple task.
You might consider an external (USB) hard drive as an alternative -- particularly if you have a newer computer that has USB-2, Firewire or ESATA ports. USB-1 might make it too slow for your purposes. If you have more than one computer, an external drive is a big advantage. You only need one. You can then use it to shuttle all your files between computers, say from your Happy Trails computer to your home computer and back.
More on hard drives.
Add a DVD burner: They're great for backups -- DVDs are on of the most durable mediums you can get. Use your burner to make DVDs (or CDs) with music, slide shows, etc. too.
Japanese Proverb: If you believe everything you read, you'd better not read.