I assume by "Europe" you're mostly talking about the Eurozone. Here are some things to look for in euro coinage.
If assembling a date set sounds a bit daunting, then a type set would be adequate, but be aware that for denominations 10 cents and above, the design of the common "map of Europe" side was changed in 2007.
Mintmarks:
This Wikipedia page lists all the mintmarks privy marks, initials etc used on circulating euro coins, but only one euro-issuing state has "mintmarks" in the conventional sense of the word: German euro coins have five possible mintmarks: A, D, F, G and J. Some of the 2002 Greek euro coins also had mintmarks, hidden inside the star closest to the date; see
this old thread.
There are all sorts of different circulating 2 euro commemoratives to look for. See
this Wikipedia page for a comprehensive list. Collecting all 89 of them would be an expensive affair, even if taken from circulation, given the high value of the euro at the moment!
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis