WOO HOO! You graduated college and managed to stay out of enough trouble that the Navy gave you a commission. While your other friends are interviewing for jobs, you're getting paid. In fact, you're probably getting paid to sit around, since you are likely in a stash job and not doing a whole lot of useful work.
You will NOT always have free time, especially as an Ensign. The minute you get to your first school or ship, kiss that free time good bye. Since you have it now, here is my top list of things to do now that will make your life infinitely easier later down the road.
1. GET A CAC READER AND INSTALL IT ON YOUR COMPUTER
Almost every DoD/DoN website you'll want to access is CAC-enabled, or soon to be CAC-enabled. The Navy Exchange sells CAC readers, or you can buy them online through Amazon or a variety of other stores. The software to install them can be found at http://www.militarycac.com/.
2. GET A SHADOW BAG (http://www.shadowbag.com/servlet/StoreFront)
I try not to promote specific manufacturers, but I wish I had one of these when I first got in the Navy. You will have a myriad of little items for your uniform that will attempt to disappear all the time. Put everything in one place so you don't lose it. The money you spend on the bag will be made up by the money you don't spend on medals/ribbons/ties/cufflinks/etc. that you end up buying at the last minute before a Navy Ball.
3. CHECK YOUR RECORD
You just got in, what could possibly be wrong? You'll learn to not ask that question soon enough. After about a month, check your record to ensure you and your education are properly entered into BUPERS.
4. GET FREQUENT FLYER NUMBERS FOR MAJOR AIRLINES
You'll fly. A lot. Even if you aren't a pilot or NFO. Go ahead and get rewards cards for major airlines. I typed up a little card with all the numbers on it, laminated it, and keep it in my wallet. It's easy to pull out and use the number when you check in your bags, and before long you'll have enough miles to take trips of your own design.
5. GET AN OFFICIAL PASSPORT
You'll be traveling as an official military member on orders to a variety of foreign countries. You need a regular passport, but I also recommend an official passport. It's given me enough leverage on more than a few occasions to avoid being booted from a full plane, and get through customs quickly so I can make connecting flights. And heck, its free. More information here: http://travel.state.gov/passport/get/first/first_836.html
6. SETUP YOUR DTS ACCOUNT, AND TAKE SOME DTS TRAINING
I don't know how many people whine about DTS. There is a nostalgia about filing paper travel claims. Let me tell you, having done both paper and DTS, I would NEVER go back. Paper claims suck. I used to wait weeks for a check. With DTS, if done right, you get paid electronically for travel within 3 days. Get your account on DTS created, and take training at Passport. The hour or two you spend learning how to create orders and file vouchers will save you countless hours later.
Get started with DTS training here: https://www.defensetravel.dod.mil/Passport/bin/Passport.html?CFID=21517872&CFTOKEN=55547232
7. GET SOFTWARE
Get used to using Microsoft Office, as that is what the DoD uses. You can get deeply discounted versions of the latest Office and Windows software, just ask your friendly IT or N6. Also, on NKO, you can download a variety of software for language learning. I highly recommend that once you figure out where you may deploy to, that you learn a bit of the local language. Not only will it help you in a liberty port, but if you learn enough of it, you can take the Defense Language Placement Test and possibly get paid for it too!
8. TRY TO FILL GAPS IN YOUR ORDERS WITH SCHOOLS
You may have large gaps of time in your orders. If you are in a fleet concentration area, like Norfolk, VA, and have a month to kill, you can write yourself no-cost TAD orders to go to a local school, instead of sitting around in a bum job that isn't going to do you one bit of good. Go to the CETARS website (on the Electronic Links page) and see if a local school has dates that work for you. If so, create no-cost orders in DTS and go! If you haven't started your PCS, you can always call the initial accessions detailer and see if they will modify your orders to send you to that school enroute.
9. WORK ON YOUR DISTANCE EDUCATION
Start doing some distance education. Whether it is through NPS, DAU, NKO, or wherever else, try to make your time at work count for something. I finished 6 joint classes and 2 acquisition courses while I was waiting to in-process at various commands. Many of these courses go on your official record and make you more competitive to promote. Since you're often at work anyway, instead of wasting your time on Facebook on the NIPRnet, use it to make yourself more likely to promote.
10. SEND YOUR TRANSCRIPTS TO NAVAL POSTGRADUATE SCHOOL
Since you're at your school and still remember where the transcript office is, go ahead and send an official transcript to NPS. It will take them some time to process it, but then you'll be in their database. NPS offers a number of masters programs, both certificate and degree, via distance learning. If you're in their system, its easy to sign up. And, if you later want to spend a nice year at sunny Monterrey, CA, getting your masters degree, you'll be able to quickly assess how much work you'll have to do, and how much more time you can spend at the beach.You can find more information at http://nps.edu/Students/Prospective/Index.html
11. Learn about House Hunting Leave
Navy house hunting leave is what most new ensigns manage to get screwed out of early. Follow the link above to my post about it. You need to read and understand the instruction now, since you move a lot in your first four years in the Navy, and it can mean the difference in being a constant basket case and settling easily.