To use a USB drive make sure your BIOS support booting from USB. If you download Vector make sure you get the live CD. AntiX, Puppy Linux, SliTaz, Lubuntu, Vector Linux are all good systems you can fit them in a CD and they don't require too much memory to run.
To make room in your hard drive for a Linux distribution you need a tool like Gparted which is included in some liveCDs then create a partition with it and install your new distribution there.
Another fun way would be find a second hand hard drive, remove the one with Windows and install Linux in it. If you want to go back to use Windows you could just swap hard drives and presto.
If you still have that other computer you said got mess up you can harvest the hard drive and use it for that purpose. Or if the issues with your other computer is just software issues you can try to bring it back to life with linux.
Good luck to you!