Want your own car, but just for a short while? Here's an option. Download the Traficar app, find an empty vehicle (their clearly-marked fleet includes 2- and 4-door sedans, as well as vans), scan the QR code and you're in. After checking the vehicle over, off you go and you can leave the car in any legal parking place. Everything is done through the app, which is easy to use and works great. That said, taxis are cheaper, and you don't have to search around for one - they come to you. Also, you can't leave the car outside city limits (but you can drive it to another Polish city like Warsaw, Wrocław or Kraków where Traficar operates). As such, this is the ideal option for big shopping trips, or maybe picking up friends at the airport, but not much else.
Note that you need to have a valid driver's license that is recognised in PL. As with regular car rental, citizens from countries that didn't ratify the Vienna Convention on Road Traffic cannot drive on their home licenses; unfortunately this includes Americans, Australians and the Chinese, among others, for whom an International Driver's License is required.