I have a Virgin Mobile Pay as you go phone, is there any chance that will work in Hungary, Italy and Germany?
Not that phone. Virgin, in the USA, uses Sprint's network. Sprint uses the CDMA standard. Europe uses a different standard, GSM.
The only US phones which can work in Europe are:
The phone should also be carrier-unlocked. Then you can buy foreign carrier SIM cards and pop them in your phone. It's cheap to use. However, unlocked phones are generally a few hundred dollars more than locked ones. If you buy a phone without a contract, it's probably unlocked.
A locked phone would still use the American phone plan. It's expensive to use abroad. But locked phones are cheaper to buy. If you buy a phone with a contract (i.e. for two years), it's probably locked.
If your phone is not GSM it won't work. You could buy a cheap prepaid phone over there for about €50→€100 or so (I'm guessing here) and use that. Might be cheaper/easier than sorting out a phone in the USA.