Oscar_Ross,
Okay. I think I've got a pretty fair idea of where you stand and what your options are. Luckily, any posts to this thread are copied to all those who have participated already. So, others can chime in, either in general support of my ideas or to suggest alternatives.
As I see it: your girlfriend got the Mac from another friend, so chances that the original OS Disks are around is close to zero.
It looks like you are the one who is going to use it. For the moment, let's assume so.
Option 1. Buy an Install Disk from Apple OnLine Store. Use it to do a clean install. This will clear out all the "rubbish" and more that has accumulated and leave you with a "new-feel" Mac. But 5 versions behind the current OS and no longer officially supported by Apple. Potential security risks the minute you go on the Internet. Need to be awfully careful. However, the clean install and clear-out can similarly be accomplished by option 2.
Option 2. As Nick said; as you are running Snow Leopard at the moment, however slow, you can click on the App Store app (found usually in the Dock, but always in the Applications Folder - Open Finder > Applications). From the App Store you can download the latest OS, called El Capitan, 10.11.2. It is free. For the download, you'll need a good broadband connection.
Once downloaded and running - then do what you started out to do - Reboot holding down Command + R. This boots you into the Recovery partition from which you can Click on Disk Utility and do a clean install, wiping everything off the HDD and leaving you with a "new-feel" Mac running the latest OS.
Problem is that 4GB of RAM, although generally regarded as the minimum, is going to make your Mac seem pretty sluggish. 8 GB would be better. You can't change your processor, so there's a certain limitation there.
The choice is yours. Quick & easy option 1, but leaving you with an outmoded OS and potential security issues. Or option 2 which brings you bang up to date; but takes more time & patience and may be a touch slow.
If it were me, I'd take option 2 and just not push the Mac too hard with heavy duty apps.
I hope that's a fair-minded summary. Others will freely chip in with their views.
Ian