/ If I move to a big city in Poland will I come across a lot of English speakers?
While from experience, I don't deny that especially in the major urban centers, e.g. Warszawa, Kraków, Poznań, Wrocław, Gdańsk etc. there are literally scores of younger Poles who are both competent in English as well as eager to practice their language skills, if anyone still thinks that outside of these areas, in less traveled places, for instance, that the run-of-the-mill person will speak fluent English, I maintain they're in for a bit of a rude shock!
Were it almost any other European country, such as Germany, Czech Republic, Scandinavia, I'd say that of course, even in remote places, there will be a number of tourist-friendly folks, equally competent in English.
I ought to have qualified that last statement! If you're plan on working in either the service or education industry, even the legal field, English alone should not pose a problem these days. Entertainment as well has by now become almost thoroughly international, and so from that perspective, there'd be no pressing need to "know" Polish (aside from the obvious).
Were you though to choose, for example. agriculture, sports, possibly medicine (but not research), a solid grounding in basic Polish might not be a bad idea:-)