By train: Kolding hasn't got a very big train yard (2 tracks), but trains between Aarhus and Copenhagen normally stop in Kolding. If not, the 2nd-largest train station of the counry in Fredericia has a train departing for Kolding every 10- or 20 minutes.
By car: When following the E45 motorway (mid-Juttish motorway), follow motorway exit 62, which should take you directly in to the city at ease.
By plane: The nearest big commercial airport near Kolding is Billund (2nd-largest of Denmark). From Billund, either take the train or a bus to Kolding (both easily found. If you can't find one, take a bus to Fredericia and then take the train to Kolding (the personnel will happily help you on your way).
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).
Many restaurants can be found in the city centre of Kolding with various themes: oriental food, pizza, traditional Danish food, American buffets and steak houses can all easily be found in the city centre and near the train station.