Although Myanmar has recently eased on its travel regulations, a journey from core Myanmar e.g. Yangoon is possible but can be quite an adventure and will require some additional paperwork as well.
On the map it looks like it would be an easy access to the Mergui Archipelago by crossing Thailand on its thinnest point directly to the western coast of Myanmar, but unfortunately in this area are rough mountains, hardly any roads on the Myanmar side as well as no boarder crossing points for western tourists. So that leaves Ranong as the only direct access point towards the Mergui Archipelago.
The easiest way is to fly in to Phuket International Airport and board a safari or sail boat either in Phuket or in Khao Lak. The border town of Ranong is a four hour car ride from Phuket Airport.
How to get there from Yangoon:
You can find regular flights from Yangoon(Rangoon) to Kawthaung(Victoria Point), which is the gateway for discovering Myeik Archipelago. It takes only about 1 hour 40 minutes to fly. There are several domestic airlines operating between the two destinations. Air Bagan, Yangon Airways and Air Mandalay are the most popular airlines offering flights from Yangoon to Kawthaung. But remember flights between these two places are known for frequent cancellation or delays due to weather and other unpredictable circumstances. From Kawthaung, you can hop on to the express boats to reach Myeik.
How to get there from Bangkok:
If you want reach Mergui from Bangkok, you can either fly or travel by bus or train to arrive at Ranong, which is a small town on the Thai side border on Pakchan River. Mergui is situated 40 nautical miles from Kawthaung and Ranong. There are regular flights from Bangkok to Surat Thani. From Surat Thani, you can reach Ranong by car, which will take about 2 hours 30 minutes. Also, you can board the train to Chumphon and from there you can travel to Ranong by car. Alternatively, you can fly to Phuket from Bangkok and reach Ranong by private or public transport.
How to get there from Phuket:
Phuket is one of the easiest way to reach Mergui Archipelago, since you have the options to travel by air as well road. There are regular direct flights between Phuket and Ranong. You can also find numerous buses and mini-vans travelling between these two destinations. After reaching Ranong, you can board the long tail boats at the jetty to get to Kawthaung. Simply ask the boat man that you need immigration stamp and he will guide you in the right direction.