Should I take a day to rest between exercise?

Exercising. If you are weight lifting, yes. You can do this by working other body parts. Back and legs one day, chest and arms the next. Good to have one or two days to rest every week. Make sure you get enough protein if body building. If cardio may train daily using decent pace.
It depends. On many factors. Including if you are just starting out or advanced, the intensity of your workouts, resistance vs cardiovascular workout, whole body or certain muscle groups and others too. One of the most important things is to listen to your body. If you are particularly sore or notice performance drops, you probably need some rest/down time.