I have ordered a Brno taxi and transport company Brno-transport from Viena airport to Brno and their service was really good. The driver spoke english, I had drink free and I could pay by a credit card. It was realy cheap and their service was provided by Mercedes. Also on the highway to Prague the driving was very safe.
I love going by car around Brno! I know it is not enviroment-friendly and stuff, but try to use a public transportation every day! You will give up sooner or later! By the by, I need to take a bus half of a journey to my work anyway, so I am not too evil, right? Enviroment loves me ;)
I used to use taxi only exceptionaly, like when I missed my night-bus at 1 o'clock. It was bloody expensive. I guess nothing has changes. Be careful with the taxi drivers. They might know English but they might also know how to cheat. No offence. So it is better to ask in advance how much you will pay. And if in doubt, always ask for a bill for later complaints!
You may get easily by car to Brno / Brünn, as it is connected with the highway to Olomouc and to Prague.
The fastest - but certainly not the most beautiful - way to Vienna might be via the highway through Bratislava.
A lot more scenic is the road through Mikulov, as you may combine it with great castles like Valtice and Lednice.
Please keep in mind that you have to pay an extra-fee for the highway :
Buy your sticker for the car at the border or at any petrol-station !!
As a motorcycle tourist I can only recommend one way to get to Brno.... by motorcycle! Seriously though, I have driven there by car only once, in 1992 and it was fine.
Getting to Brno from Prague and from Vienna (nearer than Prague!) can be done by train or by coach.
When working there (1992/93) I came home at Christmas by train to Vienna from Brno via Breclav, local train to the airport and Austrian Airlines to London. It wasn't cheap and it took all day. The train journey from Brno to Vienna took longer than the flight!
It is possible to get long distance buses/coaches to Brno from London picking up at points across Europe. I did this once and the journey took 24 hours almost non-stop with a short break every so often and a longer one in Prague.
Journey's through the western border near Rozvadov is speeded up with the building of the motorway from Prague via Plzen to the border. One the southern ring under Plzen is completed it will be considerably quicker.
Within Brno and local area: Tram and bus. Ticket strips are available from news kiosks and tobacconist shops. In the city the tram network can get you quite a way out. For outlying villages buses are frequent as well.
Local train services from Brno station can take you also to outlying places as well.
Brno is situated at the crossroads of the D1 (Prague - Brno) and D2 (Brno - Bratislava) motorways. Both these motorways form part of the trans-European East-West highways (France - Ukraine: E50) and North-South (Scandinavia - Balkans: E55, E65).