Go for it and have fun! Just make sure that you have a mode C transponder above 10k.

As someone else said, doing a spiral all the way up would take up unnecessary energy and limit your climb, so go out 10 mikes and then back and forth.

A 172 should have no issue getting to 12,500 (or higher). Also, now that it's winter you will climb even easier.

You said you're young, so hypoxia shouldn't be a big issue for you at 12,500. I've done a lot of XCs in non-pressurized planes between 11000-12,500 no problem.