I want to purchase Fallout, but I like moving between my laptop and my desktop. Is it possible, and HOW TO move the save file between installs?

The savegames are located in : Fallout_Directory slash DATA slash SAVEGAME
You can move 'em :)
note: gog forum can't display slashes.

you can do it by simply copying your savegame directory (which is X: slash Fallout directory slash DATA slash SAVEGAME) to some kind of memory card, or whatever else you use, and then pasting it to the same place on your other computer.
hope it helps)

Here's a related question: I don't have a SAVEGAME folder in my Fallout/DATA directory. I was also looking for my character sheet I saved to a text file but couldn't find it. Are these files/folders hidden?

There should be a SAVEGAME directory, with SLOTXX directories in it (XX beeing numbers).
And this folder isn't hidden.

It actually is hidden for Vista users. Try navigating to this folder:
C: (slash) Users (slash) USERNAME (slash) AppData (slash) Local (slash) VirtualStore (slash) Program Files (slash) GOG.com (slash) Fallout 2
Change USERNAME to whatever yours is. Change (slash) to actual slashes. The savegame folder should be in the Fallout 2 folder.