In my opinion (just got out earlier this year):
Boot camp: Become invisible. Don't draw attention to yourself. Follow instructions and act like you have completely embraced the Navy culture. I cannot think of anything worthwhile that you can get awarded in boot camp that will carry over into your nuclear career.
'A' school: Show up on time, never break any rules, and don't let yourself get so stressed out that you end up throwing in the towel and getting re-rated to something like parachute rigger (this happened to quite a few people I knew). A school isn't about how smart you are, it's about how you can regurgitate everything exactly how they want you to. You can volunteer for class positions to get a good evaluation (class leader, assistant class leader, etc.). This will help you make E-5 early.
Power school: Same things, but now with theory, and slightly more lax. Volunteer again to get a good eval. Oh, and never underage drink or let anyone else do it around you. This will get you kicked out faster than just about anything, besides cheating.
Prototype: Previously, you were able to sit in a chair, take tests, and be successful. Now you will have to interact with people in order to do well. To be successful, you should know the material, but you should also try and get your stuff signed off as quickly as you can. Typically, the first two or three people to qualify in your class will get good evals, and will get looked at for special duties, and staying at prototype for an extra three years. If you plan on staying in becoming a staff pick-up and staying at prototype will be the quickest way to make E-6 by the time you get to the ship, and being first in-line for Watch Supervisor quals and making Chief (E-7).
Ship: Similar to prototype, but you will have way more responsibility. Be the go-to-guy and volunteer to do every single task. If you work really hard your first couple years, you can get in good with your chain of command and get awards, promotions, and more management type jobs. If you are making it a career, I would strongly recommend looking into going officer. There are some really good programs to get into and you will basically make double what you would as enlisted. If you want to stay nuclear, do this at a training command. If you want to do something else, do it while you're on the ship.
There's so much more information that I could give you, but honestly you will just have to figure it out yourself. Don't listen to the people with bad attitudes who are just waiting for their contract to end. If you want to be successful, align yourself with people who have similar goals. Good luck.