When in Chiang Mai, you might think Chiang Rai is just around the corner — but not really. A day trip to Chiang Rai will cost you about 4 hours each way in a minivan. That means you spend most of your day on the bus. In Chiang Rai, there are a few special temples to visit; 'The White Temple' is the most famous. People also visit Chiang Rai for the 'Blue Temple'.
Chiang Rai itself is a nice town, but it is especially popular because of the beautiful mountainous surroundings. My advice would be to stay there for a few days and book accommodation in the mountains.
If you still want to do a day trip, here is an overview of day trips to Chiang Rai.