Vienna's public transport system includes the S-Bahn (suburban train), U-Bahn (subway), Trams (Strassenbahn) and Buses. On your trip to Vienna you are most likely to use the U-Bahn and Tram inside the city, and maybe the S-Bahn for getting from and to the airport.
The transport system in Vienna is unified, so when you buy a ticket, you can use any form of official transport with this ticket.
A single journey within the inner area (Innenraum) costs €1.70, and is valid for a long as you need to travel to get from one point to the next. As soon as you double back on yourself though, you will need another ticket. To avoid any confusion, you can buy a day ticket for €5.70 which includes unlimited travel for a day within the relevant zone. These tickets can also be purchased in other time blocks such as 48 hours (€10) or 72 hours (€13.60).
If you intend to do a lot of sightseeing, it might be worth buying the Vienna Card. This includes 3 days unlimited travel and reduced admission to many Viennese attractions for €18.50.
You can buy these tickets from almost any inner city station, and also on trams or buses (although this costs slightly more). Make sure that you stamp your ticket in the validating machines. If you have not validated your ticket, or if you don't have a ticket you can risk a fine of €60.
The system is barrier free, and works on the honor system, but there are patrols which come around to check your ticket every so often.
One important note is that the airport is not within the Innenraum. It is located outside the city, and requires an extra single ticket to reach.
Using a car within Vienna is generally a bad idea. If you are arriving by car, the best idea is to get as close as you can to the Ringstrasse (the road that runs around Vienna), and then find a parking garage. There are many of these just outside the ring, however they are not cheap. Once you are inside the ring however, you can get around almost exclusively on foot, as much of the area is pedestrian only.
Vienna has a compact city center, so getting around on a bike is an option. That said, many of the inner city streets are pedestrianized, and this means in addition to no cars, bikes are also not allowed here.
Throughout the city there are many bike paths to follow, and Vienna in generally is bike friendly.