I'm one of those people who likes mass transit and I've now used it all over the world. The system in Vienna ranks up there with the best, don't be afraid or hesitant to use it. It's fast and convenient and not that pricey and gets you very close to where you want to go.
A tourist will mostly be using the trams which run in the ring road. Just be aware that several numbered tram lines run on the same tracks, often following each other but then veering off in other directions. So sometimes it doesn't matter what the tram number is, and sometimes it does. Beware Rick's advice that in order to go a complete circuit of the ring you'll need to make a change at some point. This was another case in which my Streetwise map was invaluable since it shows the tram numbers and where they go. Tourbooks mention a special tram on the ring for tourists, that no longer seems to be running so just use the "civilian" version. It's fun and insightful to do the whole ring, you should do it at least once and once you're onboard, you could even just stay on and go around again.
The Vienna system works on the honor system. You don't need to pass a ticket through a turnstile or deal with a driver; it's assumed you already have a ticket or a pass and it's been validated if needed. So that puts the burden on you, dear tourist. It turns out that the validation machines are inside the trams themselves, but there may be only one per train and it may be in a totally different part of the tram - hunt it down and use it. In my experience with honor systems there will be surprise inspections that result in large fines, and tourists are not exempted. There will also be a ticket machine somewhere on the tram, or at some tram stops. Again do not hesitate, find that machine and do the deed. The machines are set up with multiple languages, always look for the little British or American flag on the screen and you'll be fine.
I also took the U Bahn out to the train station just to have a look around, again it's up to you to find and use a ticket machine. And also again, be aware that multiple numbered subways run on the same lines so do your research. Or in the case of a train to a plane, look for the little airplane graphic.