Chiang Rai is a popular enough travel destination that you won't have any trouble finding places to eat, whether you want authentic local food or a taste of some place else. Most of the restaurants are clustered around the center of town, on the streets around the night bazaar and up to the clock tower area.
Pork stir-fried with hot basil and chili, from Golden Triangle Cafe
Most of the restaurants tend towards the very casual, and of course there's plenty of street food available. For the adventurous, there's a large food court styled place in the night bazaar, with stalls selling a number of popular local foods.
White chocolate cheesecake and a hot latte from Ho! Kai Coffee.
One thing you'll find plenty of in Chiang Rai are coffee shops, and they make for nice places to sit and relax after spending a few hours on your feet wandering around. The city is near the center of the growing region for high quality arabica coffee beans, and there are at least a few shops owned by growers. The biggest concentration of shops is right around the clock tower, and within a block or so from it. In this area, I would recommend the Doi Chaang branch or the Golden Triangle Cafe. Further away from the city center, Ho! Kai sells organic coffees and some pretty good sweets as well, while the Wawee coffee shop is the most western styled, with a large comfortable indoor seating area.