Transport in Brno
Whole Southern Moravia region has it's own integrated travel system called IDS JMK. For you it means you can travel around the region (by bus or train) very cheap. Try IDOS to find your connection.
In Brno there is public transport provider called "Městská hromadná doprava" which operates with trams, buses and trolley coaches. I personally reccomend this type of transport. It's cheap and fast.
In city center there is no problem with catching a taxi. They will drive you anywhere. But I repeat, be careful, some taxi drivers could be dishonest and try to ask for more money than they could.
Parking in center of Brno is little harder but you always find some spot. If you stay in hotel, there will be probably some possibility of parking included. If you need to park your car in the center and you have no option, you can try park in front of Grand Hotel near “Hlavní nádraží“ - Main Train Station. Parking there is pretty cheap and comfort.