PM me if you want more. I'll give you a brief synopsis.
You should qualify Dolphins within 12-18 months. I don't know who told you two years, maybe he was stuck in the shipyard. That being said, operating at sea AND your work ethic will be the two greatest contributors to you qualifying quickly.
You've got two sets of quals: Back Aft (Engineering) and Forward (Everything else).
Engineering consists EOOW for at-sea and EDO for in-port. I stressed to the JOs in the wardroom that being on the in-port watch bill was more important to me than if you can qualify the forward stuff, so I greatly encouraged the new guys to qualify back aft quickly. Your wardroom should help you out on the priorities. In short, to qualify, you'll study manuals and procedures in order to get check-outs and perform practical factors. Some will be rote memorization, others you'll gloss over for understanding; its all apart of the learning process to understand what is needed where. Check-outs allow you to demonstrate knowledge in a particular subject area while practical factors are specific operations you're required to complete. All of your check-outs and practical factors will be contained in your Qualification Standard, which will spell out exactly what information and practical factors you need to complete to get qualified. Once you're complete with this, you go to a board with the Captain. On the board, the Captain, XO, ENG or whoever else feels like they need to be there, will ask you questions over various aspects of the watch you're going to be standing.
Once you get a division though, your time gets split between the division and quals, and its up to you to figure which is more important and when! Daily divisional duties will consist of reviewing maintenance schedules, attending divisional or departmental training, reviewing documents and records of maintenance, and doing audits and surveillance of maintenance. Hopefully you've got a good Chief takes you under his wing and shows you the day to day... After all, it's his opportunity to train his future department heads.
Forward watch standing culminates with Officer of the Deck (at-sea) and Ship's Duty Officer (in-port), and finally Dolphins. At-sea, it all revolves around learning about the ship's systems, service, combat, navigation, communication, etc., and learning how to utilize them as a ship driver. Again, tons more manuals and procedures on how to operate every system of the ship, how to operate the ship, and another qualification standard. In-port, the concentration is on safely approving maintenance on various ship systems. There's a few extra small qualifications along the way, PM me if you want a list. Again, finish the qualification standard and you get to board with the Captain.
Once you've qualified, you still need to learn and get better at watch standing. You'll rotate divisions (probably) so that you can learn how other divisions and departments work. At around your 2 year point, you get to go to an 8 week school (mostly self-study) for your Engineer exam.
For the sake of brevity, I'll stop, but that should be a decent amount of info.
Remember, your experience may vary =)