New Computer - Only want to restore files from backup

Hi everyone. I'm new here and joined to ask a specific question.

If you get a new computer, are you able to use the backup from your old computer to only restore specific files? If so how? Does it matter if you get a computer with a completely different OS (e.g. transferring files from a Windows 7 backup to a Mac)?

I haven't bought a new computer yet, nor has my old one died. However I just downloaded a utility to get some information about my hard drive's health, and according to the program my drive's health is looking pretty grim. The computer still performs quite well and I've had very few issues to date, however I want to be prepared for the day that my laptop meets its maker. When that day comes, I'm almost certainly going to get one with a newer OS (currently on Windows 7), so I don't really have much desire to restore my old OS or the programs I had installed because I would probably be better off reinstalling them.

Thanks!