From Thailand
It is possible to enter Myanmar at the Mae Sai border post. A 14 day visa is US$10 or 500 Baht. Note, if your US$10 is not in pristine condition, the authorities will not accept it. They really want 500 Baht instead since it is worth US$16. The visa is only good for the Tachilek-Kengtung area and is not valid for travel to other parts of Myanmar. Be warned that the Myanmar Immigration authorities will hold on to your passport for the duration of your stay in the country giving you an entry permit instead. You will get your passport back on crossing back over to Thailand. Holders of a normal one-month Myanmar visa cannot use it to enter at Mae Sai. Be sure that you have the appropriate travel documents to re-enter Thailand at Mae Sai, you will only be able to get a 15 day entry pass into Thailand rather than the usual 30 day one that you get at the airport. If you have an actual visa and want to keep it, remember to get a re-entry permit in advance.
In case you are a Thai citizen, you can make a temporary border pass at the Immigration office about 2KM before the border on Pahonyotin Rd in Mae Sai for 30 baht. All you need is your bat prachachon ID card.
By road
Shared taxis US$12 and buses US$8 run to Kengtung 4-5 hrs in the mornings. A permit is necessary but bus and taxi drivers can make the arrangements don't show up at the last minute. Onward travel from Kengtung to Mong La on the Chinese border is possible with permission from Myanmar immigration, easily organized in Kengtung.
By plane
Air Bagan and Yangon Airways fly to Heho via Kengtung daily. Air Mandalay offers a non-stop flight to Mandalay. All have connecting or continuing service to Yangon and other parts of Myanmar.