The dating customs of Poland do not differ that much from American culture. A date would proceed as it does here, the differences occur in a more traditional attitude toward how to treat a woman.
Manners in Polish dating differ slightly from American dates in that the Polish woman will expect you to open doors and allow her to go first. When meeting a woman it is acceptable to kiss the woman on her hand. When you know her a little better you may kiss her on both her cheeks. If she is carrying anything heavy it is expected that you would offer to carry it for her. Taking a more traditional approach to manners generally is the best course of action.
It is expected that a man would pay for a meal or anything else that you might do on a date. Do this the first time and she may agree to split the bill on further dates. It is not seen as attractive to talk about money and how much money or material things that you have. Polish people are frugal with money and therefore they will not expect expensive things. When arriving on a date it is customary to bring a small token gift, this could be a flower or a small box of chocolates.
Polish people are well-educated and whether they are male or female it is good to show an interest in their interests, education and/or job. Make sure that you listen to them and do not boast or be arrogant in conversation. Polish people will appreciate an understated self-assurance rather than anything showy or egotistical. It is not acceptable to drink too much; Poland has a social problem with men drinking too much.