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Lots of rules. If your wife was under public insurance before then she should be able to join again. If not and over the age of 55 then she might have to take private. If you don’t work you would get covered under her public policy automatically without cost as would any under-age children. Working you need your own policy. One usually has to qualify to be allowed to take private, public is the default. One needs a threshold salary to get private.

Look at the Wiki entry for healthcare in Germany to get a good overview. You can also post specific questions here but please look through what has already been posted. This subject has been cover ad nauseam already.

As Tom already said above, rules are complex.

Once here, you should apply (within the first three months of stay - it will apply retroactively from the day of arrival) for membership at any public insurer (the costs and conditions are very similar). Public ones are preferable in almost all situations (only exception: both of you are working, young, healthy, childless, earning well and uninterested in a socially distributing system - then you could consider the private scheme instead.

The public insurer checks your eligibility.

If you are rejected there (or want to go to private right away), you should engage a good insurance broker to select the best solution for your case among the confusing thicket of available options. (As always: Keep in mind that brokers want to earn commission from insurers, not necessarily work in your favour.)