If you are the adventurous type climb Lions Head (mountain between Signal Hill and Table mountain). Beautiful views all the way up and easy enough to do with first-time climbers. Take water and snacks though. you'll appreciate them once you summit.
Dogs are allowed on most beaches but there are a few that are dog-free if you have a phobia. Signs on the beachfront will tell you if dogs are allowed there and whether they have to be on a lead or can run free.