If you are on a tight itinerary you might travel by air, but by doing so you might be missing out on some of the memorable scenery and local color. In contract, a leisurely riverine trip would not only enable you to enjoy the scenery but provide you with a change to meet people and even witness some itinerary vignettes along the way. For along the river trip your travel agency will be able to arrange a chartered pleasure craft, or if you are adventurous by nature and really enjoy the hustle and bustle you might take a trip on the public transport.
On the other hand, if it is during the hot summer months, travelling by hired minibus, or a packaged bus tour operated by one of the travel agencies might be the best way as you would be able to travel in comfort and make many side-trips (optional itineraries) and enjoyable stops that would not be possible if you were to travel by rail. Moreover, since there are no traffic jam and air pollution problem even in the capital Yangon, motoring seems to be by far the most attractive mode of travel.
For short trips, there are besides the usual taxi-cabs, open pedicabs called side-cars locally plus pony carts in towns like Mandalay, Mawlamyine, Bagan and Bago and horse-drawn carts in Pyin Oo Lwin (Maymyo).