I have a compaq laptop that just broke, it fell and the monitor cracked, and I'm going to buy the same computer as a replacement. There are filea that I need on the broken computer, so how would I save those? Would it be safe to just put the old hard drive into the new computer?

Its unlikely you can find the same model. If you can, then yes you can swap drives. If not, I would get a drive enclosure to fit a 2.5" drive and copy the files that way. Or have the store where you buy the new laptop transfer the files to an external drive.