Can someone get married right after basic training?

During training and before graduation it would be difficult to impossible. I do not believe there is any military regulation preventing it though. The best course of action though is to wait until you reach your first duty station, unless that station is overseas. In that event, you should have a short leave time between graduation and the time you are required to report, use this time to get married. keep in mind that most of your paperwork is going to change after marriage: dependents, pay, insurance, power of attorney, etc. And, you as the service member must chase this all down.