I am also an adoptive parent, but the process for domestic adoptions is very different than that of international.
Contact a lawyer who handles private adoptions (that's what you're doing) and they should be able to give you the info you need to get started. You will need the lawyer to draw up the paperwork as the first step is always the termination of parental rights. My understanding is that private adoptions are much easier than going through "the system", but as always talk to a lawyer who specializes in private adoptions to find out.
I think the issue of medical care/etc... is up to you and the birth mother, but you will want to put everything in writing up front.