Welcome to the Navy...
While it officially kicks in when he enters basic training, you might want to give yourself a month (or at least a few days) of overlap to make sure that you're properly in DEERS (the Defense Eligibility Enrollment Reporting System) and your husband has had a chance to get you into whatever Tricare system you choose.
I suggest reading up before he goes as there are some choices to make and he isn't going to be able to call you and ask what you want.
Good luck to you.