By train: Kolding does not have a very big train yard (2 tracks), but trains to Aarhus and Copenhagen leaves at least once every hour. If not, the 2nd-largest train station of the country 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 a Airport Taxi 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).
Due to the Bosnian war in the 1990'ies there are several bus companies, mainly serving the Bosnian diaspora, which provide a cheap and clean way of getting to the other side of the European continent.
Kolding isn't very big, so getting around in the city center 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.
Because most of Kolding's sites are very centralized to the center 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 meters from the city center 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).
Kolding also has quite a big swimming facility, Slotssøbadet, which is located only a few hundred meters from Koldinghus Castle.
Many restaurants can be found in the city center of Kolding with various themes: Asian food, pizza, traditional Danish food, American buffets and steak houses can all easily be found in the city center and near the train station.
You'll never walk alone (English Pub) is the most popular pub in Kolding. You can also dinner there (burgers and indian chili). On tuesdays you can participate on Quiz Nite. On Saturdays you will have live music. On Sundays, Spanish Tapas. http://denengelskepub.dk/