You'll get some of them for destroying particular enemies in each level (always the same powerup from the same enemy), but several more are also hidden in buildings and other ground facilities too, so make sure to bring ground-weapons and destroy everything you can. Some do not appear naturally until several levels into the Campaign, but you can jump ahead with Skirmish mode (for the Reflect powerup in particular, which can get you another achievement in the process).

The three red ones must be found in Versus mode, but you can play against yourself to find them. Just keep trying different levels or terrains until you've got them all.