Honestly there really is not much you can do until you are gained by your NOSC. Stay on top of your paperwork, but let the process run its course. Respect the chain of command. Once you have orders to a unit, you can get your CAC, which is basically like having a driver's license in that you essentially shouldn't/can't go anywhere without it. You will need this for virtually everything. If you have a good recruiter, they will be working to transfer your info. Even so, there's a lot you can't do/fill out until your NOSC's INDOC weekend.
Your passports to everything are your CAC and Navy.mil account. You can't get either until you are formally gained by a NOSC and unit, as they have to sign off on the paperwork for this.