Non-rated enlisted ranks:
E-1 = recruit (airman recruit, seaman recruit, etc.)
E-2 = apprentice (airman app, seaman app, etc.)
E-3 = Airman, seaman, constructionman, fireman, hospitalman, dentalman
Rated enlisted ranks:
E-4 = Petty officer 3rd class
E-5 = Petty officer 2nd class
E-6 = Petty officer 1st class
E-7 = Chief PO
E-8 = Senior Chief PO
E-9 = Master Chief PO
("Rating" basically means job title. If you're rated, you have a specific job that you're training for. Otherwise, you're "undesignated", and you have the option at any time to "strike" for a job. You have to be rated before you can advance to E-4.)
To advance from E-1 to E-3, you really only need to put in time and fulfill a few simple requirements, like apprenticeship training and some correspondence courses, like the Military Requirements book for E-3 is required before advancing to E-3. Time requirement before making E-3 is 9 months as an E-1 and 9 months as an E-2.
To advance to E-4, you have to choose one of the jobs you're qualified for. I think the navy has 96 different job ratings--technicians, mechanics, engineers, shopkeepers, medical ratings, etc. You have to serve as an E-3 for 6 months, complete the school in your job rating and then enter the exam cycles. You can't advance without taking the exam, and you can't take the exam without fulfilling the basic requirements. The exam score is added to your job performance evaluations, and you get a few miscellaneous points for things like medals and awards and the amount of time you've served.
Making rank after that is basically the same all over again, except that each rank past E-4 has a time requirement of 3 years, and the exams themselves usually include more and more administrative and leadership questions. (E-7 and above also go through a congressional board for approval.)
There are a few programs sailors can get into to make rank faster. If you have 30 semester credits from college, you make E-2 automatically, and with 60 you make E-3. I think eagle scouts also get E-3 automatically. If you agree to serve 6 years in a technical rating and agree to attend a "C" level school, upon completion of the school you become exempt from the E-4 rating exam (You get it as soon as your time in service is fulfilled.) If you enter the "STAR" program, you can be advanced to E-5 automatically after fulfilling those specific requirements.
Here's the path my career took (I'm an E-5 right now, competing for E-6):
I had oodles of college before going to bootcamp, so I entered the navy as an E-3. I scored high enough on the ASVAB to be eligible for Electronics Technician. That's what I chose, but it's not what I wanted originally. I failed in the screening for Top Secret security clearance (I had more than 20% debt load), so I was disqualified from Crypto Technician Interpreter.
The electronics ratings are 6-year programs, so I was able to skip the normal seaman apprenticship school and go straight to electronics "A" school (the basic course every electronics technician goes through). From there they branch out. 2/3 of the students go to other ET schools, the others to into Fire Controlman schools (for very specific weapons control radars).
I went through the ET side.
All ET's and FC's go on to a "C" school after "A" school (it's in the contract), and all are exempt from the first rating exam. "A" schools are just basic electronics training; "C" schools are for specific systems. (My first one was a communications system, my second one was an airport radar.)
All told, sailors will spend anywhere from 4 months to 2 years in schools before going to their first assignment, most go to ships. Your job on the ship depends on the schools you attended. I was assigned to Radio Central because my first school was in radios. I worked there 3 years, then went to my second "C" school and was assigned to a navy air base (shore duty).
For more information, you can cruise the links at
www.bupers.navy.mil.
