If you understand Judaism and want your kids to be that way, fine. Being Jewish, bound to the TANACH (Bible) requires a tremendous amount of effort on a daily basis and for the vast majority of ignorant and lazy Jews, it's just too much.
If you marry your Jewish girlfriend, the kids will be Jewish according to Jewish law (Halacha). If you raise them that way or not is almost irrelevant, they're Jews, ignorant maybe, but still Jews.
I’m not bound by Jewish law. If I baptize my kids, they are Catholic. In fact, in order to have an inter-faith marriage, I will have to promise my future pastor that I will do the best of my abilities to raise the kids Catholic.
Unless they are baptized and raised as Christians.