I decided to end this game in a draw since I knew I was gonna screw up somewhere along the line (as shown by a bishop and a rook sacrificed for nothing throughout the game), but I'm curious: How would I have effectively finished a game like this? I have no experience in a queen v. queen endgame.

Let's ignore the glaring mistakes on both sides.

White should try to transpose to a pawn endgame. Qe4 is the main threat. Look for forking checks.

The main idea in queen endgames is to make the king safe to be able to make the pawns go forward. Q + 1 pawn vs. Q is incredibly difficult, precisely because of the many checks the defense has.

Yet, here my plan would be to bring the king around b2 where he has more or less a pawn shield and to push the f pawn. 

As Rudbidium pointed out, if you can trade the queens even at the cost of a pawn you win easily. So if you manage to make the king safe without losing a pawn, he cannot stop the f pawn without trading his queen for yours.

Yes, but it is Black to play after 39.Qxf4. He can try 39...Qa4 for example, which probably picks a pawn.

Qe4 wins.

You could've just kept on checking him until you force a trade of queens.

At the risk of repeating myself : queen endgames are incredibly hard, much harder than rook endgames for example. If you are playing someone with good tactical skill, I can guarantee you won't force a trade of queens just by chasing the king around, you just help the opponent for a 50-move rule draw claim.

No, you need to push the pawns to threaten promotion AND make the king as safe as possible, which is not always obvious.

The endgame here is indeed probably won, but as Black I probably would keep on playing until queen trade or promotion, even against a very strong player.

I agree this is most probably won. But the structure is spoiled and there is a big chance Black could pick one or two pawns by the time White's king finds shelter on the queenside, which is the reason why it does not appear such an obvious win to me.

I know the 50-move rule is without pawn moving or any capturing ; but if, as suggested in post #7, you keep checking the king with the queen, you could run into it. If you check 30 times, end with a misplaced queen, move it, Black to play has "only" to check 20 (or 19) times to get a not-so-deserved draw. This was merely an answer to that post.

I totally agree on the fact that even without knowing how to play this, you should at least try it. White has here absolutely no danger of losing (if not on clock). Even without the a and f pawns, with only the doubled on the c file, I would play it until it was "really" drawn.

As for the queen endgames odds : indeed, two pawns up queen endgames are almost all theoretically won. But winning them is another matter. The same goes for most Q+p vs Q positions, that, although they are won, aren't won easily (EDIT after post #11 : that bad phrasing means that the won Q+p vs. Q positions are not all easy, not that many Q+p vs Q are won). I didn't read Reuben Fine's book, but I own Villeneuve's  Les finales (Endgames) which is the reference in French, and queen endgames are by far the most difficult of all (although approximatively 100 pages are about rook endgames !).