Most visitors fly into and out of Bandar Seri Begawan. Brunei has direct air connections to around 30 destinations, including regional Asian capitals and a number of Australian cities. Because of the difference in exchange rates, it can be 40% cheaper to fly to Brunei from Malaysia than vice versa if you buy your ticket locally. Airport departure tax is USD 3.00 to Malaysia and Singapore and USD 7.50 to all other destinations.
There are express boats from BSB to Lawas and Limbang in Sarawak, and to the duty-free island of Labuan, off Sabah, where you can connect with boats to Kota Kinabalu. The main overland route from Sarawak is via bus from Miri, but it's also possible (although fairly expensive) to travel overland between Limbang or Lawas and Bangar, in the Temburong district of Brunei.
Transport around Brunei is by bus, hire car or taxi. The public bus system is cheap and reliable but only in and around the capital. If you want to explore the hinterland you'll probably need to hire a car, though there is a fast, regular boat service from BSB to Bangar.