Tiara Mas offers daily direct transfers to Sumbawa Besar from Mataram for Rp 70.000. Leaves 9AM, 12 noon, 3PM and 7PM. Takes 7 hours. Including ferry. Tiara Mas office Mataram: Jl. Pejanggik 108a Phone: 0370642597. Office Sumbawa Besar Jl. Yos Sudarso Depan Kodim. Phone 037121241. You can also book for Tiara Mas from the Perama office in Senggigi for 125.000 including pickup from your hotel with transfer to Mataram.
Buses from the west (Poto Tano and Taliwang) will stop on the bus terminal, far from the central area hotels and the airport. Bemos (small vans) cover the distance to the center of the town for 5,000 rp per person (extra might be charged for luggage).
There are a lot of bemos in the city. Just catch one it cost only 2.000 - 4.000 Rp.
There are 'tours and travels' that arrange transport to the West, including Mataram on Lombok (includes ferry). For addresses of the offices see the 'get in' section. They don't offer any transport to Bima. If you want to go to Bima, Lakey or Hu'u take an ocek to Sumur Payung bus terminal (8.000 - 10.000). There you can take a bus to Bima (80.000, about 9 hours, 1 stop). For Lakey change bus in Dompu.