Cześć!
This is a great question!
Each district has a public library and many neighbourhoods have their libraries too. Anyone can register for free, it's enough to bring your ID card or passport (possibly a student ID but it shouldn't be required). You can check where's the nearest public library
HERE. The rules may vary between the libraries, so it's best to inquire directly upon registration but you can usually borrow two up to four books at a time for one month, this can be, of course, extended. All public libraries have children's sections (myself, I love children's books and, in my opinion, it's a wonderful way to learn a foreign language).
If you're interested in books in foreign, your best bet might be Biblioteka Wojewódzka or the university library - Biblioteka Uniwersytetu Łódzkiego, you may also want to check the American Corner of the University of Lodz for books in English.
I hope this helps but let me know if you have any questions :) I hope you'll find a good library soon, let me know!