What would be the best way to approach painting interior red bricks?

To make sure your paint adheres and dries properly, you'll need to thoroughly clean the brick. Use a wire scrub brush to remove any dirt or dust, then apply nonsudsy trisodium phosphate (also called TSP; wear gloves and safety goggles), and wash thoroughly with a heavy-duty cleaner. Then you would have to prime the bricks with a stain blocking primer. The you can paint the bricks either with concrete paint or a good acrylic latex paint. You could also white wash the brick or get a kit

Sherwin Williams has a good tutorial on painting fireplace brick that will help for any brick.

www.sherwin-williams.com/homeowners/inspiration/project-center/how-to-paint-a-fireplace