Seems you need to learn some more about Myanmar before going there indeed, hope I can be of help.
If you want to go to Moulmein or Mawlamyine, you can take a bus from Yangon, it should be around 20000 to 25000 kyat. However, if you ask, you should definitely stop in Kyaiktio on the way. It's the place where you can admire one of the most famous Myanmar attraction, the golden rock. If you are going this way from Yangon, you should not miss it, and even if it is not in your plans, Mt. Kyaiktio and its Golden Rock is a must see. I don't remember exactly but bus fare should be around 8000 to 10000 kyat, and then from Kyaiktio (when you come from Yangon, you should actually stop at Kinpun), you can take a bus to Mawlamyine.
If you want to go Yangon, Kyaiktio and Hpa-An, you should probably have to stop in Mawlamyine for the night, but actually there isn't anything amazing about this city, I would even say it is a bit seedy, a few temples on the top of city, the sunset to be watched from the river and a slightly colonial architecture are the highlights of the city. For me it is not worse staying more than half a day.