This is actually really easy - I've been to Tivoli twice, and it's conveniently right next to the central train station.
As Wikitravel says:
It takes twelve minutes by train to get from Kastrup to the central
station (Hovedbanegården) in the city centre. You need a ticket for
three zones which can be purchased from one of the automated vending
machines or the ticket counter located inside the atrium and costs 36
Kr for a single journey. The Copenhagen Metro also connects
Kastrup with central Copenhagen, with trains leaving every four
minutes during the day and every fifteen minutes at night, taking
fourteen minutes to the city center (for the same ticket and price of
And then from the Hovedbanegården station, follow the signs across the street to Tivoli. Enjoy it!