Most important thing is to give the bricks a really good clean first. You'll want a stiff brush or broom. You'll also want to vacuum the bricks afterwards. Make sure you get rid of any loose mortar and dust. Painting is up to you but a couple of coats of undercoat (making sure you get into the cracks) would be wise before you paint with your chosen top coat colour. - Goldmember
We painted over a few red brick walls when we moved into our house. We used Taubmans 3-in-1 as it is a primer, sealer and undercoat in one. Bricks absorb paint and a normal undercoat would not give you the coverage that is required. We used brushes to get into all the cracks and then used a roller to get bigger coverage. It was a huge job that took hours, but it was worth it.
Maybe look into a spray gun as it will cut the paint time in half. Not sure if you can use the 3-in-1 in the spray gun as it's quite thick, but definitely for the top coat. We painted over the 3-in-1 with a normal interior paint from Dulux. - KingStreetReno
We were initially surprised how porous bricks can be so we used a prep and seal for the first coat and then two coats of water-based paint on top. Bricks gather dust so I’d recommend against anything that restricts your ability to wash/clean once in a while (matte finish for example). - paul_2126