The best way to get around Visoko is on your feet! It is not a very large town, and so you can reasonably get anywhere if you walk. It is also very peaceful and pleasant. Many people from Visoko enjoy just going out for a walk during the evening for no other reason than to enjoy the night air and sunset. If you have problems walking or it is hard for you, there are of course taxis. Three taxi stands that I know of are near Bijela dzamija where Carsijska meets Podvisoki, right over the bridge when you drive into the city (crossing from the autobus station), and at the autobus station itself. You can flag a taxi of course too. There is no public transportation in this small town other than that, but if you are going to another town in the region you can take Centrotrans, especially to Kakanj or town on the route Sarajevo-Visoko.
Best way to get to Visoko is taking the autobus. The Centrotrans autobus lines in Bosnia is very good, and you can take them anywhere in Europe as well. You can also take the train from Sarajevo or Zenica, but the train is still not the best option. Aggression on BiH ruined the service quite a bit.
There are very many autobuses that run from Sarajevo to Visoko. It only costs about 5-6KM for a return trip! This is easiest and most convienient way into Visoko! You can take the tram in Sarajevo to autobus station, buy your ticket there, and then it is just as easy as getting on board! The ride is very beautiful too. When you arrive at the nice autobus stanica in Visoko, you can even walk to the Hotel, as it is just across the bridge. If you have a lot of baggage, there will be taxis right at the station that can take you to the Hotel or anywhere else for not much money.