Ubuntu Linux is the way to go. I am a Linux geek, and have used it for years to put new life into old AND new computers-whether it be with 256 MB of RAM, or a person who owns a brand new gaming rig, but can't keep from downloading everything he finds online, and I can't keep his computer cleaned up from the viruses he gets!
Computer slowdowns are caused by TONS of things BESIDES viruses-registry errors (which will build up over time, I guarantee you), unused programs, startup applications galore, Windows updates (yes, Windows updates can and will slow your system down), and having bad hardware (hard drives going bad do inhibit the performance, although you shouldn't jump to conclusions yet).
However, on Ubuntu Linux, you don't have to worry about these! Viruses=nonexistent, registry errors=no registry to have errors, unused programs=easy to uninstall, startup applications=easy to disable, updates=no slow downs, and bad hardware=gives you warnings (it will tell you if you have a bad hard drive BEFORE it dies-and usually at least a month before it will die).
Anyway, the really nice thing about Ubuntu Linux is that it requires only 0.5 GB of RAM/memory, and there are plenty of desktops available-you can use the built in Unity environment, or add a new environment like KDE, GNome, Cinnamon, LXDE, XFCE, and the list just goes on and on. The best thing about it? IT'S TOTALLY FREE!
Now, to get this wonderful OS, you need to backup any data you want to keep (of course). Then, go to http://www.ubuntu.com/download/desktop . Download either the 32 bit or 64 bit, depending on what your computer uses (if it has less than 4 GB of RAM, go for the 32 bit, just to be safe-if it has 4 GB of RAM or more, use the 64 bit). Then, burn a CD. Now, all you need to do is boot into the CD, and there will be an option to Try Ubuntu Without Installing. Then, you can decide whether you want it or not. If you want it, select Install Ubuntu (don't use the Wubi installer on the Ubuntu website-if you want to keep Ubuntu for a long time, you will have to manually migrate from Wubi to a new partition, and trust me-that isn't fun, as I have had to do it).
If you need any help with Ubuntu, just post on the Ubuntu Forums-it's a very helpful place where I like to help out frequently (I am Ubun2To on the Ubuntu Forums).