All officer programs in today's navy are extremely difficult to get. Don't let that discourage you however. There are also other programs that can get you into an officer program. If you want to go to the academy, apply for an academy program, and also NAPS. NAPS is the Naval Academy Prep School, and if selected, you get one year of "prep" school, then get an appointment to the USNA. If you want NROTC, apply STA-21 and also BOOST. Completion of BOOST also enables you to potentially get a USNA appointment. Ensure you talk to the Command Career Counselor once you start training to ensure that you are aware of application deadlines and the latest requirements.
A popular misconception is that NAPS and BOOST are for minorities only. This is not true.
Its very hard to stand out and get a great recommendation at nuke school, as there are tons of applicants. Its not as hard to stand out once you get to the boat if you work hard.
Bottom line...if you want a commission, apply again, and again, and again if necessary.