In Dhaka, the only way to get around is by using one of the 300,000 traditional colourfully painted rickshaws. The city practically teems with them, and they are excellent value.
Out of the city, Biman, Bangladesh's airline offers seats quite cheaply to most of the main destinations within the country, including Chittagong and Cox's Bazaar. By land, the trains and buses cost very little, but unless you turn up very early, and fight like a wildcat, your chances of sitting down for the journey are minimal.
Otherwise, Bangladesh has a superb water transport network, which although very slow gives a real sense of the colonial grandeur of the country as you cruise along the rivers with a chilled Pimm's in your hand.
It is not possible to hire a car and self-drive in Bangladesh, although it is easy enough to hire a local "driver" but a good idea to test them before you commit.