Young miss, I appreciate your heart in motivating young minds but it doesn't sound like you have thought everything through. The fact that you have a job now is germane to the discussion. If you move, you are going to be in a different state with different standards and levels of regulation. It would behoove you to find out the most restrictive standard and seek that level of education/certification prior to moving in with him. I don't know what that is, but my instinct tells me an associates would not provide you with the flexibility you need. I could be wrong but it sounds like you are assuming that the way things are in your hometown are the way things are everywhere.
Additionally, you will be applying for a new job requiring people to trust you with children with no local references and your plan is to have little or no formal education for being a pre-k/k teacher. His time spent moving around in the early pipeline is going to make it difficult for you to get employed because he will only be in Charleston for a maximum of a little over a year; this fact will also make it difficult for you to seek a degree, since there are no one-year programs. It can take as long as a year to even get invited to an interview depending on what the job market looks like in the place (and child-care is a very common military spouse profession, making it a competitive market for you in most places). Getting a more advanced degree also provides you with the flexibility to take employment elsewhere and earn money until a daycare position opens. If your relationship is strong enough to get married, it is strong enough for you both to pursue your respective educations until he gets assigned to his first sea tour.
Another thing to consider is that BAH as a married E-4/E-5 may or may not cover the rent of a 2 bedroom apartment; it depends on the duty station and where you are willing to concede to live. If you have children and want to rent a 2-3 BR house it most likely will not cover that cost. Many junior enlisted with children live in base housing, but I can't say that I have heard very good reviews about that arrangement from most Sailors, while the old salts will tell them they don't know how good they have it. If the plan is for him to do 6-and-out and wait on children, then that gives a lot more financial flexibility to you. But if he's thinking about the possibility of a career as a nuke, then finances are something you guys need to discuss.
As for the logistics of marrying your boyfriend post swearing in, I already told you that his training pipeline is rigid and will not give him time off to do it. The qual program for nukes is also rigid -- his CoC will not be able to extend his qual goals by 2 weeks to get married because they don't set the timeline, NR does. If they give him time off on the boat to get married then he will have to make up the time later as he will go dinq.
I'm not trying to make it sound like it's all gloom and doom; I am married myself. My CoC gave me a week off to go on a honeymoon in a drydock period. I wish I could tell you that was guaranteed, but it's not. On the employment front, my wife always happened to get a call right around the time we were ready to PCS, not at all helped by the fact my boat did a change of homeport mid-tour to go into an engineered overhaul. She was making a little over $20/hour in a hospital admin job before we got married and had offers in her home city for 1.5-2x that salary. Hospitals are everywhere so getting employment should be easy, right? Wrong. Unfortunately, the two local hospitals weren't hiring. When she looks at classifieds, the overwhelming amount of admin/clerical type jobs require a master's degree and 5 years of experience. She did some substitute teaching and considered daycare, but no one was hiring full-time positions; part-time at $10-15/hr is all that was available and not worth the cost of a babysitter.
Hopefully you have better luck than we did, but the reality is that flexibility is the name of the game if you wish to be a working military spouse. My wife has difficulty finding employment because outside of working a hospital admin job, she is ill-qualified to do much else. I knew this when I married her, though, and once my children get a little older it's likely she will go back to school. The reason I post this is so that you and your boyfriend consider these things before you jump into marriage; there are a LOT of divorces in the military and it typically stems from having misguided expectations from the beginning paired with financial troubles.
On the admin side, he routes a chit to notify his CoC that he's getting married. They can't order him not to tie the knot; this is to make sure his CoC is informed and can take action to support him. Once you are married he brings the certificate to PSD and updates his page 2 and SGLI beneficiaries if desired. Then he enrolls you in DEERs and if you desire Tricare prime he fills out a form for that. Once you two have a lease or mortgage, he brings it to PSD to get BAH. It's fairly simple and all this would take an hour of his time, not including time spent waiting in line.
The Navy will not pay for you to move in with him; they will pay for you to move in every subsequent accompanied PCS move.