I'm an 18 year old guy and I started riding when I was 17. I own a CBR 600 (fuel injected), worked for it and bought it myself. With all honesty, I have never dropped it nor have I ever been in an accident. Also, I have never felt that the power was too much for me. Yes, it CAN do 0-60 in 2.8 seconds, BUT that does NOT mean that every time you touch the throttle that you are going to fly down the road at lightning fast speeds!
People that fear 600cc bikes seem to think that if you touch the throttle that it is suddenly going to lose control and fly out from underneath you like some raging bull, this simply is not true! If you are an idiot and show-off then you could easily kill yourself on a 125cc moped, but if you keep your mind about you and have a reasonable capacity to be mature and learn (which from your question it looks like you do) then you should have no problems learning on a 600.
I took the MSF course and passed with 100%; I didn't even miss a single point! What most people don't realize or don't tell others is that 600cc bikes are actually quite tame (and kind of panzies!) below 6000 rpm. 600cc bikes are designed to have a spiking power band that really takes off after you hit 6000 rpm, which in normal every day driving I usually shift around 5000 rpm and if you even push to 6000 rpm you are already moving so it is quite controllable.
If you buy a 250cc bike (like the ninja 250r) then it will just be a complete waste of time and money because you will be tired of it after 2 months. If for some reason you cannot learn on a 600 then you can always get a twin cylinder sport bike like the Kawasaki Ninja 650R which still looks like a super sport but not as much power as a 600cc and won't get boring like a 250cc.
Honestly, it is on a case by case basis. If you are mature and will actually take your time starting out instead of being over-zealous and going full out throttle then you will be fine with a 600cc bike. Some people cannot handle a 600, some people can. It depends on a mix of things such as your ability to adapt and learn, maturity, and confidence. Take a moment to honestly answer this question: "Are you mature and will you honestly take your time to learn?" If the answer is yes then you likely can handle a 600, if the answer is no then you should not be on the road until that answer changes. I suggest you take the MSF course FIRST, see how you do, and then decide what to do after that. If you score well and feel comfortable during the course then you can likely handle a 600.
Good Luck and Happy Riding!