By bus from Chiang Rai or Chiang Mai. Buses from Chiang Rai are very frequent (every 10-15 min), cost 40 Baht (one way) and take about 1.5 hours, while buses from Chiang Mai take about 4 hours (210 baht one way). All public buses terminate at a bus station well south of Mae Sai, so you have to continue onward to the city proper with red songthaews waiting outside (20 baht per passenger fixed fare). The blue songthaews to/from the Golden Triangle and Chiang Saen, on the other hand, leave from a stand in the city centre a few hundred metres down the street from the border. The last blue songthaew that departs for Chiang Saen leaves at 2PM.
There are also day-tours taking visitors to Mae Sai, the Golden Triangle, and a couple of other places to see on the way as well. Tourist minibuses are more expensive (around 800 baht for a day tour from Chiang Mai including lunch) and, if fully loaded (but this is not always the case), are much less comfortable. However, this is still an option worth considering, if you want to combine your visa run with a sightseeing tour through Chiang Rai province.
There are several minibus services which run between Chiang Mai and Mae Sai, tickets for which can be purchased at most tourist and travel agencies in Chiang Mai. Prices are approximately 700 THB for the round trip which takes 5 hours each way, with a 1-2 hour stopover mid day as people cross the border for shopping and re-entry stamps.
Mae Sai is quite an underestimated town as many visitors just breeze past it on their way to a Myanmar borderrun. But if you would like to visit an authentic Thai border town there is much to see here for you.
The Tam Pla caves, about 12 KM from Sae Sai are absolutely worth a visit. Take a torch (flashlight) as there is no artificial lighting in the caves.
Visit the caves along the main road.
Wat Phra That Wai Dao, (before Immigration, turn left and go through covered bazaar). Built on a small hill up a steep staircase, overlooking Mae Sai and Tachileik, this temple and its stupa are unremarkable, but there are a few interesting monuments around it. Next to the stupa is a small Burmese temple that gives a nice taste of what awaits on the other side. Towards the river is a multistoried monument to King Naresuan, a Lanna king famous for beating back several Burmese invasions and dispatching the Burmese crown prince in a duel, and just in case the message of this isn't clear enough, there's also a giant scorpion statue brandishing its claws towards Tachileik.Free. edit
Take a picture with the gate marking the northern-most point of Thailand
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  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.
Kik Kok restaurant(no seriously, there is a sign outside!)
It is about 1Km up from the border on the right as you walk in that direction and there are a few Thai restaurants about this distance along the main road away from the tourist shops near the border. It closes about 8pm.
The one immediately next door, run by Chinese descent people will give you a good feed quite cheaply even later, with alcohol if you want it, and the menus are easy to work on. They are either in English or with pictures of what you get.
Anantapura. This pleasant restaurant is probably the classiest in town. On a hillside, on a road that runs parallel to, and is about 1 km west of, the main Pahonyothin Rd, the views down the narrow valley are stunning with the Shwedegon chedi accross the border in Myanmar lit up at night. Be sure to try the smoked duck curry. Service can be slow but this is not a restaurant to come to if you are in a hurry.
Doi Wao Temple. In front of Doi Wao Temple (The scorpion temple just above the market stalls of Chinatown) there are two food stalls selling excellent noodle dishes. One, run by a short middle-aged plump woman, sells Khao Fuen (Shan style made with blocks of tofu and in a soup or a yam-spicy salad-style) and the other is run by an older skinny Thai Lue woman usually wearing a wide straw hat who sells excellent nam ngiew noodles Yunnan style and very good Khao Soi Thai Lue style.
There are several comfortable coffee shops on the main road leading to the immigration checkpoint. There are some nice bars (of which the Monkey Island bar is definitely one) and some nightclubs in town. Ask the locals for the nightclubs, they know best where to go on which night.
EQ Home, (From the main street and just before the Myanmar border (while facing the border), take a left through the shopping area. About 500 meters from the main street the hotel is located at the Y-crossing.), ☎ 085-0325539, . checkout: 12:00. Clean and cheap. Not the Hilton but very friendly atmosphere and great location. Dormroom is 4-dorm with personal curtains and satellite TV. Hot water. Towels, blankets, soap and unlimited free drinking water included. Just a very short stroll away from the Frog Island bar. All business English spoken. Perfectly good motorbikes and bicycles available for rent.Dorm 100, Fanroom (shared bathroom) 150, AC room with TV (private bathroom) 250.. edit
Thip Sukon House Hotel. This hotel is probably the best of the lot on the same road as the old King Kobra hotel.From the 3rd floor you have a view right into Myanmar and the Sai River.
Piyaporn Place Hotel. 77/1 Moo1 Weiangphangkhun. (No website - #2 on Trip Advisor) This standard Thai three star hotel is right on the main road about a 10 minute walk from the border. The clean well presented rooms contain all the usual amenities of a hotel in this class. The loud maid phones on each floor which resonate down the corridor may put some off. Ask for a room at the opposite end of the corridor. Standard rooms go for 800 baht except in the core high season of December and January when rates can reach 1,200 baht. Asian buffet breakfast included.