Kolding isn't very big, so getting around in the city centre is quite easy. If you wish to go to more remote locations, e.g. the botanical garden of Geografisk Have, buses are cheap and drive all over town. It should also be possible to get a taxi cheaply (by Danish standards. Taxi-driving is mostly very expensive in Denmark.)
Because most of Kolding's sites are very centralized to the centre of the city (Church of St. Nicholas, Koldinghus, Geografisk Have), most travellers won't have to do much travelling once the city's been reached.