Bangladesh is a small country: it has only a little over 140,000 square kilometres in its area, of which over 10,000 square kilometres are actually allocated to water bodies. Getting around Bangladesh is thus fairly simple, not least because of its geography and the multiplicity of rivers and streams that wend their way through the land, making possible a very healthy system of waterways that supplements the land-based routes already in place. There is a well-developed air travel system in place as well, perhaps because of the state’s focus thus far in its tourism promotion efforts: most of the campaigns run by this fledgling state for tourism have been primarily centred on domestic tourism, which means getting around Bangladesh by plane (or any other way, for that matter) is quite affordable as well as easy.
The first option is obviously that of getting around Bangladesh on land. There are at least four ways of doing this: by car, by cab, by bus, and by train. Of these, the train is probably the best bet, since the train system in Bangladesh is fairly cheap and reliable. It is not going to be terrifically clean, of course, so expect some wear to the seats once you get on.
If you do plan to use an automobile for your travels here, be warned that Bangladeshi drivers are usually a law unto themselves: it would be far safer for any tourist to just hire a chauffeur or trustworthy driver for their rented vehicle than to drive it themselves. For those who cannot afford such expenses, there is always the bus, although buses in Bangladesh are the equivalent of trains in Mumbai. In other words, expect buses to be fairly worn-looking and constantly full. It would be wiser to opt to ride only in the more expensive buses, which are often identified by the fact that they are air-conditioned as well as far less stuffed than the normal buses.
Now as for cabs in Bangladesh, be warned that they are not the cabs you would find in New York or London: a cab in Bangladesh is an auto-rickshaw (or a rickshaw sans the “auto”, which would be a manually pedalled/powered one). These are fairly cheap and quite convenient… if you have the guts to get into them and brave the driving antics of the operators. They are not recommended for the faint of heart or cautious because of the reckless driving often observed from their drivers.
Getting around the country by air is possible by taking any of the various domestic flights. There are just under 20 airports in the country and all major divisions have at least one (as do popular tourist destinations), which makes plane travel awfully convenient. It is also quite affordable, so it may be the method of choice for many persons. To find out if flights can serve your needs during your travels, go to the capital of the division of the country you are in: nearly all the airports are located in these areas.
Finally, getting around Bangladesh by boat is also possible because of the ferry services and water routes that run all throughout the country. If you are thinking of making any journeys passing from the waterways of Dhaka to Khulna, investigate the Rocket Steamer service: it is a fantastic ferry ride that shall give you the chance to see some superb scenery.