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EU regulations for roaming fees make it no longer necessary to get local SIM cards in each country.
Since you're arriving in Hungary I suggest getting your SIM card there. This guide (from late 2018) has very detailed information on getting SIM cards both at the airport and in the city.
For a start, you can check Telenor and Vodafone (in English). In theory, all packages should be valid across the EU but I still suggest to double-confirm this and check the fine print.