Bangkok to Myanmar by RoadMyanmar to Bangkok by road
If not, by 2019 in accordance with the agreement on international trade between Myanmar and Thailand. You can take a taxi across the Myawady frontier from there or rent a Songthaew, as they are a little less expensive and more regular - only a few cabs have a license to travel across the frontier, as cabs in Mae Sot are still very new, but that will probably be changed over the years.
Or, you could ask virtually any Myanmar automobile that is willing to take you with you - a tip, don't be timid if there are others already in the automobile, but try to dial using either one of the four license numbers (these are license numbers for local transportation ), because getting a license plate is a personal automobile, although it's no issue if someone is willing to take you with them.
Those cars are hanging near the river crossing the river near the river bed. Alternatively, cross the viaduct (most aliens do) and then find the transportation to Yangon on the other side - most folks take a shared cab to Hpa - and if you want to economize your fare, get on a coach to Yangon.
In the mornings there are busses and there are also many mini-vans (at least to Hpa-an). Driving times on the road from Bangkok to Mae Sot: Journey from Mae Sot coach terminal to the Thai side of the bridge: Thai side frontier formalities: 2 minutes, provided there is no waiting line.
Myanmar page: about 5 minutes, incl. letter of the registration card, again under the assumption that there are no waiting lines of travellers. Myawady to Hpa-an, 3-3. From Hpa-an to Yangon: 5-6 by road, about 6 by coach. Alternatively, to conserve some of your precious travel hours and still see the Myanmar landscape, take a Nok Air flight to Mae Sot (4 times a day) and then head to Yangon ashore.
In the past there were also Nok Airlines services from Mae Sot to Yangon, but not for the now. Chartering Bangkok to Yangon by plane is also an optional extra, although I suppose you would rather go ashore. From Bangkok to Yangon via Phu Nam Ron/Htee Khee and Dawei is also possible, but will take much longer.
First, you have to take a mini-van to Kanchanaburi (from the Siegesdenkmal for example, although there are other places of departures such as Central Pinklao) and then a Phu Nam Ron (border bus), these only run until 1 pm in the mornings? Then a very slowly winding railway goes up to Ye, where you take another railway via Mawlamyine to Yangon.
Mawlamyine offers a much quicker way to get there. Surely the Mae Sot/Myawady passage is much quicker, less expensive and there are more transport possibilities. After all, I see no reason whatsoever why recreational trips with a medication permit are not allowed. Usually I go to Myanmar on a commercial permit and this is not a big deal.
Usually the opposite is the case (e.g. with a touristic permit, although the first visitor has no other option). As you are only planning 3 week of a 90 days holiday in Myanmar, you will be well.