What do I need to talk with my recruiter about to see if my wife and kids can be moved to live with me after boot camp?

PVT (Join to see) there is a regulation that states how long a school is , makes it a Permanent Change of Station PCS move, but TRADOC policy determines when your family may join. Your recruiter has the info or can find out. You need to tell the recruiter what your situation is so they can advise you. Recommend calling or emailing this question. If something needs to be done at MEPS like issuing family orders, the recruiter can facilitate.

25S is an 18 week AIT, you aren't authorized to move your family to the training location, and if you do, you won't be authorized to stay with them. You can move them once you get orders to your first duty assignment

Even if your AIT is 26 weeks or longer, you will get very little time for a home life cause you are still a trainee.