Entry to Denmark
Denmark is a member of the European Union, and travel to and from Denmark is governed by EU law. Therefore, different entry requirements may apply depending on where you're travelling from, and what your nationality is.
EU citizens can travel freely to Denmark; citizens of other countries may require a visa. If you travel to Denmark from outside of the Schengen agreement area, you may also require a visa. See here if you are from a country where you will need a visa to enter Denmark.
Getting to Odense
From Copenhagen Airport:
Take the train from Copenhagen Airport to Copenhagen Central Station where you will have to change in order to get to Odense.
All west-bound trains from Copenhagen Airport stop at the Central Station.
There is a direct train from the Airport to Odense every hour.
Tickets can be purchased at www.rejseplanen.dk and www.dsb.dk. It is also possible to buy tickets from the vending machines and ticket offices at Copenhagen Airport. Please note that tickets are not sold onboard the trains.
Trains to Svendborg depart every half an hour from Odense Station.
All trains coming from Copenhagen with a final destination on Jutland (Aalborg, Aarhus, Esbjerg, Fredericia) stop at Odense and Middelfart Station as well.
From Billund Airport:
Take a bus to Vejle train station (Bus 43) and change to a train to Odense.
Tickets can be purchased at www.rejseplanen.dk and www.dsb.dk.
Please note that tickets can be purchased from the bus driver but are not sold onboard the trains.
The trains to Odense stop at Middlefart Station as well.