Is it good for a 13 year old boy to exercise (weight training and cardio)?

Sure, but...: Of course a 13 year old should exercise, and exercise for all of us should contain both "weight" and "cardio". This should not even be considered exercise, but a part of good lifestyle. Now if you are asking about formal, heavy duty weight training, it can be done by age 13 but should be under the supervision of someone used to working with growing teens to avoid overuse which can occur with teens.