Healthcare in Hungary is generally of a high standard. If you're working in Hungary under a local employment contract you should have access to free or subsidised medical care through the Health Insurance Fund. You should also be able to get access to public healthcare with your European Health Insurance Card, which is valid within all EU member states.
You can read more about this and what you need to quality for access to free public healthcare in our Healthcare in Hungary page.
All the best with your relocation!
The healthcare system is actually really good in Hungary. Just sort out your insurance. There also a good selection of private doctors available in Hungary.
And good luck!