The best way to travel between the islands, is with the plane. The distance is to long for going by boat unless you are very fond of boating.
I don't know how accurate this schedule is, you better check with Air Kiribati's office in Bikenibeu at the phone no listed below or at the Bonriki airport. The travel agency in the shopping center in Bairiki sells tickets, so do local agents on the islands. Just ask the locals where to find an agent :-)
We travelled by boat from Tarawa to Abaiang. It is a combined cargo / passanger vessel, with no comfort. But who cares, we were not there for the comfort! A part of the trip we were accompanied by dolphines, quite another experience to se them in the middle of the Pacific Ocean than in an aquarium!
The ship leaves from Betio harbour on Tarawa when the cargo is loaded and the tide is right, you must expect to wait, and there is no waiting room. Bring enough drinking water, it might get hot. It would be wise to bring some food as well, since there is only a small kiosk in the harbour area, and nothing to be bought on board.
On the outer islands there are no buses and few, if any private cars. But there are trucks, and it is normally not a problem to catch a ride.