Gili Trawangan is connected to Bali (Padang Bai and Benoa harbour) and Nusa Lembongan by fast boat, to Lombok via fastboat and slowboat (latter to Bangsal town) and to neighbouring Gili Meno and Gili Air by both public boat and chartered boat.
To/from Gili Meno
The public boat leaves at 16:00, costs 20,000 rupiah per person and takes 30 minutes. You can charter a boat for the same run for 200,000 rupiah. The public boat will drop you at Meno harbour, while the private charter will drop you wherever you want.
A slow boat service runs roughly on the hour or when full. IGNORE ALL TOUTS on the mainland. Fare is 10,000 rupiah, payable at the kiosk right by the beach. Trip takes 30 to 45 minutes. Private charters can be organised for 180,000 rupiah for the boat.
A bunch of speedboat services are available from Lombok to Gilis from a couple of bays south of the main port. It's difficult to justify the expense unless you're a DJ running late for a gig, but they'll whisk you over in 15 minutes.
Operators of the fast boats between Bali and the Gili Islands have a very patchy safety record and the boats are not run to a safety level you may expect in your home country. When we travel between Bali and the Gilis, we take the slow boat to Lembar.
Cidomo or horse and carts are the primary mode of transport around Gili Trawanagan. There is no motorised transport on the island.
Fares are expensive, with a ride from the port to the far side of the island or the northern tip going for about 50,000 to 70,000 rupiah. Fares increase if you have a lot of luggage.
Note that there is a whole network of trails across the island, so to get to the other side they'll go through the middle of the island rather than around it.
If you're planning on doing much exploring, hire a bicycle -- it beats walking up and down the strip all day.
Standard street bicycles are available for hire from around 30,000 rupiah a day and those with child seats affixed to the back for 40,000 rupiah a day. As with any kind of bike hire, check it thoroughly before hiring and if you're going to leave it unattended for any period of time (snorkelling or at night for example) lock it.