Marshrutka
From Didube, leaving every 1-2 hours, starting at 9 am, last one is around 5 pm. The ride costs around 7 Lari and takes 2,5 hours. Usually the driver makes a short brake before climbing the serpentines to Gudauri. Good souvenier shopping there.

Bus
Also leaves from Didube 2 times a day, one around 3 pm. Is a little bit cheaper then Marshrutka, but also slower. It took us once 5 hours to get there.

Taxi
If 4-5 people travel together it might be a good idea to share a Taxi. The big advantage: you can make stops on the way up, for example at Ananauri or at one of the numerous lookouts. According to some Georgian friends of ours the way to Kazbegi is best part about the stay ;)

Own car
If you have your own transport you can easily drive there, Kazbegi is one of the best accessible mountain areas in Georgia. One suggestion: drive at day time. The views are fantastic and the road quite challenging at times. There is no gas station in Kazbegi, but there is one in Arsha, about 6 km before you enter Kazbegi.