Is it safe to groom my cat?

As long as you're getting brushes and shampoos designed for cats, then everything should be a-okay.

If you're a groomer and have experience, then yes. But otherwise, I would shy away from it. I've seen too many cats that come in with lacerations from owners trying to cut out mats over the years. Of course, they feel terrible when that happens, and they never it on purpose, but cat skin is very delicate, and if you've got matting, safely getting it out can be a real challenge. Even in the hands of a professional groomer, I've seen the skin accidentally cut. I recommend not trying it. I think brushing it is great, but avoid using scissors or clippers. Unless you're very comfortable with it, I recommend professional grooming.