There are multiple ways to backup files
-Transfer the Files to Media such as CD's/DVD's
-Transfer the Files to a External Hard Drive Or USB Flash Drive
Are the kind of files you want to backup, just basic files such as Folders/Documents/pictures etc etc? If so you can use one of the above ways to backup the files.
Windows vista has built in CD/DVD burning software, so if you insert a blank CDR/W or DVD/RW Disc into your Burner (Assuming you have a burner) then you should have the option to Write files to the Disc, by dragging them onto the CD Drive that you have the Disk In and then choosing the Option on the left (Cant exactly remember what it is called) that says Burn Files to Disk or somthing like that. If you can open up your windows vista help and support and type in thje word "Burn" you should pull up the necessary help files.
Transfering to a External Hard drive or USB Drive is pretty straight forward. Just plug in the drive, and once Windows Vista is finished recgonizing the drive then you can just open up the drive like you would any other and drag or copy files onto it.
**please note However** You said your system is very infected. I would assume you mean infected with a virus or other malicious program. So this means that your chances of transfering that malicious program into your backup process Could be Very High depending on what you are infected with. You will need to make sure that once you Reformat you get a Good Solid Real time Antivirus and Anti Malware program installed. then BEFORE you start installing all of your programs again, i would insert the Media (external HD, USB drive or CD/DVD) into your system and then Run Full Scans on that media to make sure you didn't have any virus's or other attached to any of the files you just copied.