As a Polish citizen, do I need a passport to travel from Poland to any EU country?
You need a passport OR an ID card (dowód osobisty, but only the newer type, a plastic card, those older types of "dowód osbisty", a green book, are not valid anymore).
If you travel within the Schengen zone (most EU countries except for UK, Ireland and maybe some more, not sure, check Wikipedia or some other sources), you won't need any document to cross the border, because the are no customs controls, actually no controls, but you will still need some kind of official document within any of the Schengen zone member countries.
For the period of European Championship in football (currently in Austria and Switzerland, ending on 29th June) Austria has normal controls on its borders (Austria belongs to the Schengen zone, but it has no effect during those few weeks).
