A very common day-trip from Chiang Rai or Chiang Mai is Mae Sai, or the whole 'Golden Triangle Loop' of Mae Sai to Sop Ruak to Chiang Saen and back. Mae Sai is advertised as the 'northern-most point in Thailand.'
There is not much to see in the town. Its mainly a place to shop and look across the small stream which separates Thailand and Burma (Myanmar). Unless there is one of the occasional flare-ups along the border, westerners can cross the small bridge into Tachileik, Burma on the other side of the stream. However, there's not much to do in Tachileik except shop and you're only give a one-day visa to enter. This is not a place where you can enter into Burma for onward tourism.
If you want to spend more time in the area, there are a number of hotels and resorts in the city as well as in the surrounding countryside.