Answer:
There are a number of prerequisites before you can join, such as a written test, and various physical tests.
To start though you should contact your local Navy Recruiting Office, or visit the Navy's website (link below).
This website will give you a lot of information on the Navy, but I also recommend doing research to find out exactly what you would like to do in the Navy. From the website you can search for a recruiter in your area. They are generally available Monday through Friday 0800-1700 and beyond (8am until 5pm). Some recruiters work extended hours, to include weekends and holidays. If they don't answer they may be out meeting with possible recruits or at recruiting events. You can leave a message or try back.
After you have contacted the recruiter they will have a lot of paperwork for you to do to determine your eligibility. Afterwards you will be required to take the Armed Services Vocational Aptitude Battery (ASVAB) exam. This is a written test they administer at your local Military Entrance Processing Station (MEPS). It is a way for them to evaluate your knowledge and reasoning abilities to determine which Rating or Navy Enlisted Classification (NEC) you qualify for.
Afterwards they will do a brief background investigation to include interviews asking about drug use, mental health issues, incarceration and legal issues.
They will do a full physical checking to make sure you are in good mental, physical and psychological health.
If all of the above is fine you will move on. If there are any issues you may have to apply for a waiver. (There is a waiver for almost anything in the Navy, so don't be discouraged)
After all of that is complete you will move on to a Classification Adviser, who will discuss your job opportunities and set up your enlistment contract.