OP, what time of year are you going? I would lean to Luang Prabang, and particularly if you will be going in May-Sept, basically rainy season (like Siem Reap in Cambodia, LP is a lot more crowded with visitors in the winter/dry season). Chiang Rai is not so much of a destination but a provincial town and staging base. There are some interesting circuits to do in far north Thailand and especially if you are willing to self-drive. (or hire a car with driver). Doi Mae Tung, the Princess Mother's House and Garden, Mae Salong and the tea plantations, Thaton and rafting, in the other direction Chiang Saen, the Golden Triangle (tourist trap sign, but everybody does it anyway), plenty of little villages, etc.

For a poster above: The border crossing excursion from Mae Sai to Tachilek, Burma, is not however, one of those priority must-dos out of Chiang Rai. If you want to do it, you need to get stamped out a Thai immigration, pay your fee and get stamped in at Burmese immigration (may have to leave your passport, not sure what the latest protocol is), wander around Tachilek, and return. You don't need a tour guide for this. The main tricky part is finding the correct location on the Thai side to get stamped out (and back in) at. If Burma is in a good mood and they are allowing the 15 day pass that allows you to go as far as Kengtung before returning, then GO GO GO if you have the time. Kengtung is a fascinating area (Shan majority) with lots of hilltribes that still live and dress like they used to in Thailand 40-50 years ago. You really need to have about 5 free days from the Tachilek border, to round-trip and do this excursion justice, though.

Remember that if you return to Thailand by land and do not have a Thai visa in your passport good for another entry, you will only receive a visa-free entry stamp good for 15 days, not 30.