There's really only one place to visit in Mae Sai, but many visitors breeze past it on their way to Myanmar.
Wat Phra That Wai Dao
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.
Take a picture with the gate marking the northern-most point of Thailand