Sultan Muhammad Kaharuddin III Airport (SWQ) is Sumbawa's airport and is right downtown. You can walk there from your hotel in about three minutes if you are at one of the airport hotels or about 30 minutes from the hotels on Jalan Hasanuddin. An ojek will cost 5,000 rupiah. The airport is currently served by Garuda, Wings Air and Trans Nusa. Garuda and Wings fly to Praya (Lombok) and onto Denpasar (Bali) daily, while Trans Nusa flies to Bali direct, but not every day. There is no departure tax and tickets can be purchased at the terminal or beforehand online or via a travel agent.
Sumbawa's bus station is around five kilometres out of town to the northwest on the road to Poto Tano. Buses from elsewhere will drop you here -- they do not go through the centre of town as Sumbawa Besar has a bypass road to the west that they use. An ojek from the bus station into town (they wait opposite the bus terminal) should cost 5,000-10,000 rupiah depending on where you are going.
As Sumbawa Besar is on the main trans-Sumbawa highway there are frequent through buses; buses to Poto Tano (for Lombok) and Dompu and Bima pass through throughout the day. At Sumbasa Besar bus station we were quoted the following prices by the friendly bus office crew. Departure times were vague and variable. We'd suggest getting there in the morning to see what is happening on the day. Expect only through buses in the late afternoon, which you may or may not be able to get a seat on.
Bima: Throughout the day, 80,000 rupiah.
Dompu: Throughout the day, 60,000 rupiah.
Poto Tano: Throughout the day, 60,000 rupiah.
Taliwang: Every 20 minutes in the morning, 35,000 rupiah.
Jereweh (for Jelenga) and Maluk: Twice a day, both in the morning, 70,000 rupiah.
Mataram (Lombok): Between 06:00 and 11:00, 100,000 rupiah
Every man and his dog doubles as an ojek in Sumbawa Besar, but the town is small enough to walk around to all the main points of interest, with the exception of the bus station. There is no formal motorbike hire shop in town but ask at your hotel to see what they can rustle up -- with some persistence you'll find someone with a brother and a motorbike who is happy to hang out bike-less for a few days if you pay for the bike upfront. Expect to pay 50,000 to 60,000 rupiah per day.