While you can do pushups and pullups every day, you might benefit more from an exhaustion workout and then rest days.
One of my favorite workouts comes from the amazingly annoying Tony Horton. Pick 6 types of pushups with a number and 6 types of pullups with a number, then do 2 sets of each for a total of 24 sets. So for instance, a 30-15 would be 12 sets of 30 pushups and 12 sets of 15 pullups. Pick ambitious numbers and don't be afraid to go to muscle failure.