As Henrik says, taking a taxi would be the easiest and most comfortable solution, since you won’t have to walk or carry your luggage anywhere at all, basically. But public transportation should also be quite easy, and a fair bit cheaper, because Danish taxis are not cheap. Going by standard taxi rates, a taxi will probably set you back something like DKK 150–175 (€20–23.50, $22–26) for a three-kilometre ride that will take about four minutes.
I haven’t gone on the specific bus routes that go by Billund Airport, but Danish buses (particularly ones that pass larger airports) generally have a space set aside for prams, luggage and such things, usually on the right side of the bus between the front door (entry, next to the driver) and the middle door (often exit-only), so you don’t need to worry about having to cram your bags up on to the seat with you.
Google Maps is actually quite accurate when it comes to public transit in Denmark – it pulls data from the same sources as the purely Danish site Rejseplanen, so the results are usually the same.
Looking at your specific time and the clarification as to which exact hotel you’re staying at, this Google Maps direction guide shows that you can take buses 43, 143 or 166 – they all stop at a bus stop very usefully named Åstvej v. Hotel Legoland (‘Åst Road by Hotel Legoland’), from where it’s about a 200-metre walk to the hotel itself (Google Maps says to follow the road, then turn left, then left again, but you can cut across the parking lot, that’s quicker).
Åstvej is the second stop from the airport on all three lines (the first stop is the main entrance to LEGOLAND and Lalandia), but you should keep a lookout, since buses won’t stop if there are no passengers waiting by the bus stop and no one has pressed the ‘Stop’ button on the bus. The bus journey itself takes about five minutes. To be on the safe side, you can tell the driver that’s where you’re going, and they’ll usually make sure to stop and announce it when you get there. ‘Åstvej’ is pronounced [ˈɔːˀstvaɪ̯ˀ], in an English-based phonetic respelling approximately oast-vye, but if you just say the hotel name, the driver will probably know which stop you mean.
The company operating the buses is Sydtrafik, who have an English version of their website with more details. You can buy the ticket on the bus, but they only take cash (no cards) in Danish kroner or Euros, and you can only expect change up to the nearest hundred kroner. They also have an app available in Apple’s app store and in Google Play (both in English); you can use a payment card to buy the ticket ahead of time in the app if you prefer.
The price for a single adult (age 16+) ticket is DKK 26 (€3.50, $3.80); a child (age < 16) ticket is half price. One adult traveller can bring up to two children under 12 on their ticket for free, while a paying child (age 12–15) can bring one child under 12 on their ticket for free. So if you are travelling alone with two children under the age of 12, you would only need to buy one adult ticket for yourself, and the kids can travel on that for free.
