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.)
As Kolding is not very big, you should somewhat easily be able to get around by foot.
Church of St.Nicholas (St.Nikolaj Kirke) - one of the oldest churches 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.
Koldinghus is only a few hundred metres from the city centre and the church (they form a sort of triangle between them). The botanical garden of Geografisk Have does require either a short bus ride or walk (appx. 5km or 3 miles).