Do you know if there are differnt kinds of dermatologists like that just deal with acne and rashes and moles but not skin cancer?

No. The specialty will encounter all types of disorders of the skin. Dermatologists are all trained to examine biopsy and treat lesions. That said some specialize only in cancer surgery most notably MOHS surgery involving skin cancers with resection and closure of the wound. Not all dermatologists perform this surgery.