Regarding "must sees", what are your interests and what age range are you in? https:/…tourist-frontpage is a good place to start.
As for getting around, you can rent a car. When going from Zealand to Funen/Jutland you can go via the Great Belt Bridge, where you have to pay a road toll https:/…toll-charges
Alternatively, you can take the fast ferry between Sjællands Odde and Jutland. It offers a nice break from driving, but if you plan on going from Copenhagen to Billund, or vice versa, it's a detour. Check https://www.molslinjen.dk
Alternatively, you can use public transportation. DSB drives the InterCity trains and many Regional trains in Denmark, and tickets can be bought on dsb.dk/en. There are no trains to Billund, but you can take a train to Vejle and take a bus from there. You can plan your journey on journeyplanner.dk.
Long distance buses are also available, check flixbus.com, kombardoexpressen.dk or sortbillet.dk.
Lastly, you can fly from Copenhagen to Billund or Aalborg, but as distances aren't that long I wouldn't do it myself.Edited: June 10, 2018, 4:28 am