Accession /ak-sešn/ is the formal word being used in English for the "vstup" into the European Union (EU) by ten new countries on May 1st this year. Other English words which might be used are "joining" and "entry", but it depends on the specific sentence they are used in, and they may be used more often as verbs (to join and to enter).

There is also a formal verb "to accede" /ak-síd/. So the Slovak Republic is joining/entering the EU on May 1st. No need for "to" or "into", unless you want to say that Slovakia is acceding to the EU, or that Slovakia's entry into the EU will soon be completed. It is also possible to say that Slovakia is becoming a member of the EU. I use the "is joining/entering/becoming a member" form because I am referring to an event in the future which we know is fully agreed and arranged, and we know when it is due to happen. There is no equivalent in English for the informal Slovak expression: "Ideme do EÚ", because in fact Slovakia is a European country already, and we don't have to "go" anywhere to join the EU.