What size hard drive would you suggest?
As big as you can aford.
Can I use the new hard drive separately?
Yes you can run from it and store/backup to it.
Can I fit it myself?
An easy job as long as you have the room.
Just open your pc up and check that you have a drive bay free, and wether you want to fit it to the IDE ribbon cable or take a look to see if you have a SATA connection on the motherboard, if not why not consider an external H/D these days they are large capacitys and allso reasonable in price.
Just remember to set the new H/D to slave or cable select.