The typical home desktop is used for a variety of tasks, none of which are particularly demanding. If your PC is typically used to do word processing and web surfing, is sometimes used to play movies or basic 2D games, and is never used to play 3D games or run demanding programs like graphics editing software or video encoding software, this is the upgrade path you'll want to take.
Every Home PC should have at least two gigabytes of RAM, and if you do not have that minimum, I would perform that upgrade first. It is incredibly easy to do, as a RAM upgrade involves nothing more dramatic than slipping the RAM into the slot and securing it with pop-in fasteners (details and pictures here), and RAM is very cheap - two gigabytes shouldn't cost you much over $20 dollars at an online retailer. Once you have two gigabytes, the upgrade path for RAM becomes more muddled, because the typical 32-bit operating system won't recognize a full four gigabytes of RAM. Resist the urge to over-upgrade just because you have the cash, but Vista uses a lot of memory so its users in particular may benefit from 4GB, even if less than the full amount is used.
The CPU will be next on your list. You must first check to see what CPUs are compatible with your motherboard. Once you've done so, I'd generally recommend upgrading to a mid-range dual-core CPU. Anything over $100 dollars is likely faster than you need, but otherwise you should simply buy whatever you feel you can afford. Athlon X2s and dual-core Celerons are a good bet.
Lastly, the hard drive comes into play. Hard drive space is very cheap these days, and virtually any mechanical drive from a quality brand like Seagate or Western Digital will be a good choice. Just make sure that you have a free SATA port (explained earlier in the series) available on your motherboard, and make sure that the hard drive itself is a SATA model. Keep away from Solid-State drives or 10000RPM mechanical drives. They're meant for situations where super-qucik access times and massive read/write bandwidth is needed, and their prices are high.