Cross the bridge into Tachileik, Myanmar (which the Thais call Tha Khi Lek ท่าขี้เหล็ก). An entry permit valid for up to 14 days costs 500 baht or 10 US$. Please note the official will either try to get the 500 baht (as it is worth more) or try to charge you more. The official price (as listed in the office) is $10. Travellers using this option are given a paper entry permit and their passports are held at the immigration office until they return to Thailand. If the crossing into Tachilek is done purely to exit and then re-enter Thailand, one can forgo the paper pass and walk across the street to pick up one's passport, and walk directly back across the bridge and re-enter Thailand.
Stop at the entry point and talk with the Thai officials to organise this. It is easily done. In the Tachilek market just over the border, expect to be assaulted by any number of persons offering cheap cigarettes and viagra/cialis. Most vendors in the markets are actually Chinese and not Myanmar people, and most manufactured goods come from China.
From here, one can travel as far as Kengtung (Thai Chiang Tung) provided you hire a tour guide, 160km away, but to travel to the rest of Myanmar, a visa in advance is needed. Transit travelers can arrange for a visa and can have their passport sent to their port of exit. Some local NGOs [1] are worth visiting to see some of their humanitarian work.
Note that one cannot extend a visa at Mae Sai/Tachilek as there are no Thai consular services. Can only get a non-visa exemption pass (14 or 30 days, depending on nationality) or activate a visa already obtained.
Rent a scooter and drive through the mountains. Take the first left just before the border and just keep driving - it is one of the most scenic drives you will find in Thailand. Fuel up as you can expect a 100km ride if you ride the whole loop back to Mae Sai.
Visit the fish cave about 10 KM from Mae Sai.