The simplest way to get to the Rakhine region is to fly from Yangon to Sittwe. As with all domestic flight schedules these can be erratic so try to give yourself some extra time on either end for connections.
Renting a private boat and guide to get up the Kaladan River to Mrauk U and points further is not too expensive and makes for a very interesting trip. Note that as of Fall 2008 boats can not leave after about 11am due to possible dangers on the river if you get stuck there with a breakdown after sunset, so you may need to overnight in Sittwe after you fly in before you can head up river.
There is some concern about being on the Kaladan River after sunset, purportedly due to danger of being robbed.