To travel around Bratislava is indeed easy, as the city has a genuine network of public transportation. Although you will not find metro here, you will discover that the Old Town of Bratislava is a compact and pedestrian. Actually, you might just walk around, without using any public transport, especially if you book one of the centrally located hotels.
Nevertheless, if you need to use public transportation, the first rule is to purchase the ticket in advance at a newspaper/tobacco kiosk or machine, as it is not possible to buy the ticket from the driver. The second important rule is to stamp the ticket immediately after embarking the vehicle, otherwise it is not valid. There are random ticket inspections and if you fail to prove your valid ticket, it does not really help to claim that you are a foreign visitor. And finally, not different from other cities in Europe, please beware of occasional pickpockets.
Tickets are valid for an indicated time, you can either buy a ticket valid for 15 or 60 minutes (for 70 or 90 Cents, the 60 minutes ticket is valid for 90 minutes during the weekend and holidays), day, two, three or seven days.
Trams are the most reliable mean of transportation, as the railroad is separate from roads for other vehicles and thus trams do not get stuck in traffic jams as happens with buses and trolley buses. The tram will comfortably bring you to the city center and to outer city districts. You might enjoy traveling by a trolleybus, which is less usual in some countries, so why not take it to the Castle Hill or Slavin Memorial?
Taxis are a special chapter like in every county, so the best is to use a credible local company. It is always cheaper to call and order the taxi by phone in advance than to stop one on the street, as the fare with phone reservation is 50% cheaper. The best is to ask your Slovak friends or at the hotel reception to call the cab for you.