The first thing you need to know if you are looking to save money on computer purchases is this: upgrading is cheaper overall. Computers are highly modular; so you can often buy just the component that will get the performance you need. Upgrading isn't always the best way to do things, depending on what you are upgrading from and to. You'll need to:
You'll need to know how to perform the upgrade in question, or get someone else to do it. This series is on how to build a PC, start to finish, but the individual articles each focus on the installation or connection of a particular component. Head to the appropriate article for instructions on your particular upgrade.
If your current PC is not worth upgrading, you will need a new one. Building a PC (instructions for which are in the series just linked to) allows savings up front, and, with some good choices, you can leave yourself better routes to upgrade, over a ready made PC. Choosing parts and having a PC built will cost you a modest service fee and a little research, but you know what you are getting; and, perhaps more importantly for a cheap PC, what you aren't. Any small bricks-and-mortar store will be happy to provide advice, and many online retailers have Configuration tools to guide you through the process.
Still, if you don't know a Gigabyte from a Cheesybite, and want to get "a computer" and leave it at that, here are 3 cheap PCs that can fit the bill. If you are building your own PC or choosing parts for it, the next page contains recommendations for the different components you'll have to choose.