Everything will be new to us: first baby, first time in Poland and Warsaw.
Here is a hospital which comes high on the charts szpitalzelazna.pl/en/patients-guide
I have no personal experience with the place other than visiting. I know three couples who gave birth at this hospital and all reviewed high. As far as I understand it is state or partially state you need to do your own research. As for Damians our daughter was born there a good few years ago and if I am correct it cost the insurance company 10,000 zlotys. It was the best at the time as for the market now I am not up to date.