There are several buses daily from Myawaddy to Hpa-an (Pa-an) and Mawlamyine (Moulmein), however a Myanmar or Thai passport or ID card is required.
Foreigners can only cross the border into Myanmar at this point on foot (unusually, foreigners are not even permitted to cross with bicycles).
From the Thai town of Mae Sot, take a songthaew or other transport to the border crossing at the Thai-Myanmar Friendship Bridge over the Moei River; Myawaddy is immediately on the other side of the bridge.
Foreigners must deposit their passports at the Myanmar immigration checkpoint. The officers ask for 500 baht, whereas officially the fee is $10. They will haggle and mess around, using a variety of ruses, such as your US dollar bills being printed in the wrong year. To save (their) face, carry a selection of notes, or just be patient - the fee is $10. No fees are payable at the Thai checkpoint. Normal exit and entry stamps are issued at both checkpoints.
Currently (March 2010) it seems to be impossible to pay $10 insteard of 500 baht. When asking for an explanation of why the border in Mae Sai charges $10 and the border in Mae Sot 500 Baht (which is clearly more than $10) guards may claim that these are different border crossings and there are more foreigners in Mae Sai so it makes it easier to exchange the dollars, or something like this. In any case, just be prepared to pay 500 baht. Even if you pretend to have no thai currency with you, the officers then let you enter the town without taking your dollars but tell you to change them into baht there and pay 500 baht on your way back!
As of late March 2010, on your re-entry back into Thailand you are given visa for only fifteen days, not "30 days" (as a certain guidebook wrongly says).
While in Myawaddy the Thai Baht is universally accepted, as such, there is no need to change your Baht into the Kyat (the Burmese currency).