What's the best way to paint the brick facade of a home or business?

I would use a paint sprayer or a long paint roller to best get this project done.

https://www.youtube.com/watch?v=mnm07u6B3Tg

Two ways to paint the bricks using the same technique depending on whether you want to paint the mortar (grout) lines or leave them the natural color.

1) If you don't want it to look spray painted and want to keep the mortar lines you will want to use a roller. BUT in this instance make sure you don't paint the mortar lines. Do not put too much paint on the roller because then some will be certain to drip onto the mortar lines. Depending on the look you want with the bricks you may have go over and dab paint into the "holes" that the paint on the roller will not cover on the face of the brick. (Experiment with roller nap size. The rougher the brick the thicker the nap)

2) If you want to paint the mortar lines and still make it look like it has not been spray-painted I would suggest you paint the mortar lines first with a different color than you use to paint the bricks themselves. Perhaps slightly darker or lighter (usually lighter will look more real) than the brick color depending on the color you use on the bricks. In this case don't worry if some paint gets into the mortar lines because you can always use a small brush to repaint the lines that have the brick color on them.

Remember that most exterior bricks have a "ledge" all of the way around the brick because of the way the mortar has been applied and you may have to deal with that by hand painting that area of the bricks. If you are painting the mortar lines you will most likely want to paint them last, though to make the painting job easier you could paint the mortar lines along with the brick and then go back and paint them with the final color.

I would ask at the paint store if you need to use a primer as your first coat of paint on both brick and/or mortar. A lot might depend on where you live and what the weather is like. When painting it is always better to be safe than have to redo. You might want to get several opinions and go with the one you feel gave you the best answer (not necessarily the one YOU want to hear)