Skinny teen; push ups or weights?

I am 17 and 5'2" ,weigh about 125 lbs. Since my height issue is being essed by my doc, I was about to try weight training to get my muscles in order. I have like the smallest wrists of any other male. I mean my skinniness is such to the point of my head looking oversized. My doc tells me that my high metabolism would impede any sort of weight training, but I was wondering if either doing push ups or small weights for building my arms would help. And what would help with the abdominal area? If so, what kind of exercise for each? help. please give me a thorough answer. =)