I was diagnosed with staph on the side of my breast. I'm on bactrim (sulfamethoxazole and trimethoprim) antibiotic will this heal and how long does it take to heal?

Depends. Did you have a culture of the wound? If a culture showed infection with mrsa, it may take several weeks to eradicate the infection.
Antibiotics. Bactrim (sulfamethoxazole and trimethoprim) is used to treat a type of staph called MRSA and 10 to 14 days of therapy often suffice. Your physician will guide you.