There are 4 things: security clearance, immigration, military, and SSA.
-Security clearance: just report your marriage to your security manager and chain of command. They should know what needs to be done, if anything. When I got married I had recently re-done my security clearance in which I listed everything for my then-fiancee and reported her as such, so I didn't do anything after we were married. Since then I've had an additional periodic re-investigation and there were no problems.
-Immigration: You should know that if she is not currently in the US on some type of visa, and then you get married and try to get a marriage visa, that it takes about a year from application to acquire the visa and that she cannot stay in the US with you during this time. I did the fiancee visa which is the quickest way to get your fiancee over here (but also most expensive) and then get married in the US and adjust status to permanent resident. During the adjustment of status she is to remain in the US. See visajourney.com and uscis.gov.
-Military: Once you are married, go to the ID card office. Bring her passport and marriage certificate. She should have a visa in her passport which will be OK for the 2nd form of picture ID. She does not necessarily need a social security number at this time. Go by the tricare office if you want to sign up for Tricare Prime, otherwise you're on Tricare Standard by default. You can also sign up for Tricare Dental at this time if you want to. They will take ~$12 out of your monthly pay on your LES for dental premiums. Update your life insurance beneficiaries (SGLI) and your record of emergency data (VRED).
-Social Security: If you do a marriage visa, SSA may give you an SSN as part of that process, but if they don't go to the social security office and sign up for a SSN for her. This took me 3 visits and a month and a half since they're basically unfamiliar with the fiancee visa and there were some issues with the name check. The issue was her visa was in her maiden name, then we were married so her name changed and everything from that point was in her new name except that her passport still showed her old name.
I'm pretty sure that's about it. The immigration process was the height of bureaucratic foolishness, but they didn't give me any problems. The time from application for the visa to the actual visa issuance was 5-6 months. She came over, we were married within a few days and then we applied for adjustment of status so she could get her green card. I taught her how to drive and she got her license within a few months. It cost me more than $3000 for her entire process, the majority being USCIS application fees. Other associated costs are embassy fees, doctor's fees, immunization costs, document fees, etc. She's from Japan.