Dawei (Tavoy) is the capital of Tanintharyi Division, in Southeastern Myanmar.
There is one daily train from Yangon to Dawei via Mawlamyine and Ye, which takes just over 24 hours. It departs Yangon at 18:25, departs Mawlamyine at 04:30, departs Ye (where you change to a different train) at 10:25 and arrives in Dawei at 19:00 (at least according to the schedule - delays are common).
Ferries to Myeik and Kawthoung leave from about 35km (21 miles or about one hour's drive) south of Dawei at around 04:00. You will need to be at the jetty by 03:00 as they will leave early if possible. It takes around 4 hours to Myeik (USD25-30) and about 10 hours to Kawthoung.
Dream Emperor Hotel is in a good, central location and has simple but comfortable rooms for MYK22,000 (twin or double). There are several other hotels within walking distance.