There are two main ways to get there:
Billund is situated in the middle of Jutland, this means that there are roads in every direction. Usually tourists coming from other countries will drive along the E45 motorway. If you are coming from south, drive to Kolding and continue towards Billund. If you are coming from north or east turn to Billund at Vejle.
The airport runs two express shuttle buses to Vejle (30 minutes, 80 DKK) and Århus (1½ hours, 180 DKK via Horsens & Skanderborg) several times per day. There are also direct public buses from Billund to Vejle and Grindsted (lines 244 & 907X), Kolding (406, 1 hour) and Horsens (117, 2 hours). All of these buses stops at the airport.
Billund airport  is one of Denmark's biggest airports. Cimber Sterling Airlines  is based at Billund and flies to and from Billund to 15 destinations. The airport is served by many budget- and charter airlines, there are also flights by some national carriers like Finnair , SAS Scandinavian Airlines  and KLM , Ryanair also now offers cheap flights to Billund.
Not many restaurants are located in Billund. So go have your dinner in Vejle 30 km away where a lot of fine resturants are located although they are a little expensive they are well worth the trip, especially Trattoria Dei Re which serves tradional Italian cusine, and Hviids Vinstue in Fredericia gade with a nice French/Danish cuisine. The alternative is Legolands fastfood resturants, a Hotel Legoland.