Should I do push-ups at the age of 17?
At age 17 (assuming you're male), your testosterone levels are increasing by the day. What better way to take advantage of this than by directing the energy toward good use?
Pushups are a compound exercises. A compound exercise works multiple muscle groups. In this case, pushups exercise mainly your pectorals and triceps, and also deltoids and latissimus dorsi.
By doing pushups every day, you'll begin to not only build upper body muscle, but also train your body to be active, therefore demanding more testosterone production.
I would recommend challenging yourself. Start by doing as many pushups as you can in one go. Take note of this number. Every day, try to beat this number of pushups. When you reach a new record, keep trying to beat that record.
Ultimately, pushups will become a part of your day. You'll love doing them and you'll love the benefits you'll receive. 17 is a great age to start strength training.
Best of luck!