How do you find out about taxes, health care, etc as a dual citizen? (Hungary-US)

This is a rather unexpectedly weird situation. I’m a dual US-Hungarian citizen, and I’ve lived in the US all my life (except for short times in Hungary), and so as a non-resident I haven’t been involved in these things in Hungary. But now I’m moving back there for a year and I want to find out about both what I’m responsible to do (do I need to pay into social security? taxes?) and what the benefits might be (do I get health care? do I need to pay for it?). I’ve called the Hungarian Consulate and Embassy in the US, and they won’t give me any information on this, telling me to Google it (they deal with passports, visas, etc). My Hungarian language skills are intermediate, and my internet searching has been fairly fruitless. Any ideas? This seems like it should be strait-forward to figure out, and there should be some office that would tell you these things. I’d prefer not to pay a private organization or lawyer for advice/information on this.

Thanks for any ideas!