Is it possible to move to another country (further West) without a visa?

As a general rule, once you wish to travel to another country you should contact the relevant embassy of the country you wish to enter to obtain a visa. The consent of the Commander of the Border Guard for 15 days does not entitle you to travel in other EU countries.

However, if you entered Poland on a visa/residence permit/visa-free regime (Ukraine), you can travel to other EU countries for 90 days within a 180-day period. 

You are entitled to the same authorisation if you hold a diia.pl – an  electronic residence permit provided you have a valid travel document.