I think Schonnbrunn is a must see, in my opinion. Note that the Roman Ruins are not actual Roman ruins, but rather, is a sculpture built in 1778 (still pretty cool). The grounds are stunning, the palace is gorgeous, the views from the top of the Gloriette of Vienna are fantastic. I'd plan to get there when it opens, as it is quite popular. It's a very easy metro ride outside of the city center (take the Ubahn to the Schonnbrunn stop). Plan to spend at least a couple of hours there.
After that, I think it really depends on what your interests are. Vienna has many excellent museums. Both the Kunsthistoriches (art history) and the Naturhistoriches (natural history) museums are excellent. The Hofburg museum/apartments/treasury (apartment is maybe the greatest understatement of all-time) are great (but this could possibly be a little much in the way of ornate palaces if you are doing Schonnbrunn too).
Definitely see St. Stephen's Cathedral, it's historically significant, and Rick's book does a great job of detailing the high points inside.
If you can see a show at the Opera (we were unable to), I would do it if this interests you. Alternatively, take a tour of it, which we did. It can be tricky to get tickets - check the schedule to the left of the operahouse (when facing the front) for when when tickets go on sale, and get in line 30 minutes early.
Another thing I'd highly recommend is getting out of the city center and enjoying an evening at a heuriger for dinner. These are family run wineries/restaurants on the outskirts of Vienna proper. Most have ample outdoor seating and often feature live music. The wine is great, the food is great - follow Rick's advice on picking one. We went to two in our three nights in Vienna!
If you aren't sick of palaces, Belvedere Palace is really beautiful too, we really enjoyed it.
If you like food markets, check out the Naschmarkt (southwest of Karlsplatz) - this is an interesting way to spend an hour or so, and plenty of good places to get lunch.
This isn't our thing, but the Lipizzaner horse show and the Vienna boys choir are typically popular.
Last, with only three days, I would skip Bratislava. Nothing against the city, it has its own charms, but devoting only 2 days to Vienna doesn't do it justice.