I've almost made it to the end in Chrono Trigger on ZSNES now, but now I've got a new computer and I kinda want to keep playing the same file on that one instead;
is there any way to move the ROM to that computer, or perhaps use passwords or something?
It is completely unproblematic, and doesn't require any other knowledge than how to drag and drop files in windows explorer.
Your ROM file is (usually) named nameofthegame.smc, while the save data for the same game is named nameofthegame.srt.
If you use savestates in your emulator (meaning saving where the game does not normally allow you to), these filenames vary from emulator to emulator. I could look up what they are, but you could just keep it simple by saving the game in a way that the game intends you to do, and then move the files i mentioned earlier.