I was looking at the Veloster Turbo as well but only because of the turbo and fuel economy 26/38 and runs on regular, not premium fuel. I wanted it as a commuting to work kind of car and perhaps long trips. I predict gas to hit over $5 eventually and remain there, which is really weighing on my decision for my next vehicle. Even thought about hybrids. As I got older, safety and reliability became more important over performance and appearances. What killed the deal for me with the Veloster Turbo was the visibility issue. It seems to have some bad blind spots, I don't care how you adjust your mirrors, I prefer to see what's around me. The ability for a car to avoid an accident is to me a safer car. When I was 18, I just wanted a car that had 4 wheels and a steering wheel. Looks like for you, money is not really an issue but I'm not sure why you have narrowed it down to these two cars, they have nothing in common except they are both new debut cars, and not exactly tried and tested, especially the Veloster Turbo which I'm still waiting to see a test drive review just cause I like following news on cars and technology in general. I think the frs/brz would make for a good first car at your age, it's new and up to date with all the safety requirements. If you've already driven your dad's toys (porsche, AMG) then 200hp isn't something you can't handle. Everyone keeps saying the cars not "fast" but it's not "slow" either so seems just right then for a beginner enthusiast. Why get an old car for? You don't know how reliable it's going to be or what problems you might end up having, especially being a first time owner. Learn the rights and wrongs with this new car, how to break it in, and take care of it so that it takes care of you.
I personally would take the FRS/BRZ over the Veloster Turbo if money or gas wasn't an issue, and besides, you're on this forum so we are kind of biased already.
http://www.thecarconnection.com/news/1075718_six-month-road-test-hyundai-veloster-visibility-challenges