Actually I think there are several people on this board who think of themselves as the Scuba Police, but I digress...
As mentioned, the OW depth limit is 60 feet. What happens if you go deeper? Well, since you got an OW fill, the air becomes toxic at 60.3 fsw, until you learn special breathing techniques in AOW. Ok, I am kidding. Nothing really happens.
It is good to stay shallow at first, until you gain confidence and skill in your newly discovered habitat. There is plenty to see at 30 feet, you won't be bored, and nobody will think you are a sissy if that is as deep as ever go.
That being said in NC there is some awesome wreck diving deeper than 60 feet. If you are into that sort of thing work your self up to it slowly, they will wait for you.
Most important thing is to know your limits and approach them with respect, and have FUN.
Just curious, but didn't your instructor tell you about the 60 foot recommendation in class?