Where are the save data files for all 3 games. I thought they were the .DAT files but I guess I was wrong. How can I back up my save files?

They are in the directory where the executable is.
For example in TR1: C:\Program Files (x86)\Gog.com\Tomb Raider 1 2 3\Tomb Raider 1\TOMBRAID. The files are SAVEGAME.0 through SAVEGAME.10 and so on.

Got it! In Windows 7 it was in a slightly different location from what you said but I managed to find it. Thanks a bunch!

Would have been nice if you had posted the path after finding out where the savegames are]where the savegames are. Reminded me of that one xkcd comic ;) Here's where I found them in Windows 7:

C:\Users\<YourNameHere>l\AppData\Local\VirtualStore\Program Files (x86)\GOG.com\Tomb Raider 1 2 3\Tomb Raider 3

So I've figured something out with the windows 10 version.Since it doesn't have any TOMBRAID folder,All you have to do is basically put your saves in the main directory of the game folder and it should load like nohing.You can even edit the save to put them in any save slot you want hence(putting save.12 to save.0 will obviously put it in the very 1st slot).