Do you actually have cats or dogs?

Do you have carpets, rugs, runners or any upholstered furniture?

You'll probably see the most benefit by getting rid of whatever carpeted surfaces you can, and diligently vacuuming those you can't (frequently with a good vacuum). You might also consider a good vacuum of any blankets, or similar in the bedroom, as well as the mattress itself.

I don't have any personal-use experience with any stand-alone air filters or purifiers, but be wary of what you're buying and of the reviews you're reading. I know that some of the filters a friend used to sell for a few hundred were sourced from China for something like $20-40 landed - they were nothing special or great really (info from him), but his customers did give really good reviews.