The Yeoman won't be able to do anything about your pregnancy except add the child as a dependent when they are born, and help counsel/start entitlements as they arise. FYI, unless you personally have insurance, all your medical checkups and everything associated with your pregnancy will be out of pocket or through your insurance. Once that baby is born, your Boyfriend will add them to DEERs and get them in TRICARE (military insurance). Depending on how long you have been together and if you have discussed marriage before I would be very cautious in making a decision. I got married young in our relationship due to pregnancy, ended up with Two beautiful, healthy and amazing children... and a Divorce. Not saying all relationships end like that that begin like that but that was just mine. If you have been together and discussed it and just needed a reason... here it is! If you don't want to do the courtroom thing, military chaplains can do a ceremonial wedding for free. Nothing fancy but if you're religious it will be in their church on base.
As far as logistics are concerned, it sounds like you will have already had the baby before he goes to A school, so the Coast Guard will move your child and their belongings to where ever he gets stationed after A school. You can move as well, but you will not get any entitlements. If you fly, your baby's ticket will be covered but not yours.. etc. Unless of course you are married.
I think you're number one concern should be on your relationship and financial stability. A school is a huge milestone in an Enlisted Member's career and if your boyfriend wants to stay in and make it a career do not put in his ear for him to postpone it. Aviation rates already have a slow advancement process and delaying this would be worse. When he is at A school, I would go live with family to help out, my ex wife did (I went to a school while she was pregnant).
If you have any questions or concerns about entitlements or anything.. I am a Yeoman