Unless things have changed:
After ELT school you will go to sea or be selected for ELT-Staff Pick Up.
If you go subs you have to qualify Submarine Warfare withing 9-12 mos (no exceptions-usually).
If you go carrier you will need to qualify Surface Warfare within a few years (I'll let a skimmer expand on this one)
Once at sea you will qualify your senior in-rate watchstation: Engine Room Supervisor
Now here comes the other schools part:
While an MM3 or MM2, basically a new guy, you may have the oppurtunity to go to specialized equipment schools for various auxillary equipment, such as:
1. Refrigeration/EPA R-114 cert
2. Distillation System (8K, 12K, etc)
3. HPAC/LPAC
4. Advanced Valve Maintenance
5. Operational Water Chemistry (A pre-Engineer school for Officers, but sometimes they let ELTs go, too)
6. Basic Quality Assurance
There are also Leadership Schools, Advanced Damage Control and Firefighting, etc.
Now if you go Trident Submarines all these schools are done during the Off-Crew (Pre Deployment Training). The billets are shared amongst the other boats and amongst your shipmates, but generally they are first come first serve so ping on your school coordinator during deployment to be ready for school requests as soon as the boat gets back to port.
Now if you go fast attack (according to my buddy from the Hyman G.) you basically are at the mercy of your manpower, schedule, and who is qualified what, meaning unless you are in port, you have to be AUGMENTED off the boat for any of these schools while your crew is deployed.
If you go Carrier I don't know, but I assume it is similar to fast attack, but I also assume there are more people to share the load if a few guys are off in school.
After this you will go into EWS Quals (Engineering Watch Supervisor).
This all takes roughly 1.75--3 years once reporting to the boat depending on OPTEMPO, Ship's Schedule, and most importantly, you.
Now for the schools between commands:
These are special schools if you will be performing a special function at sea/shore-
1. QA if you are going to a QA Billet ashore
2. 3M if you are going to be a 3M coordinator at sea (Chief and above usually)
3. Recruiting School
4. Advanced Radiological Controls if you are going to a shore RADCON billet (this is not just for ELTs, but all nukes)
There is probably more, but that is all I can remember.
It all boils down to:
1. Get to ship
2. Get qualified Warfare and Senior in Rate
3. Then worry about additional schools, unless you are Trident then utilize OFFCREW to go to school since you have roughly 6-12 weeks to train, might as well go to school since you can't do boat quals unless you are sent to another boat at sea to qualify. Oh yeah, if the oppurtunity is ever present to go to sea, while you are in port:
DO IT, unless you have kid or something, but if you are just a NUB single nuke, allow yourself to be short cycled back to sea to qualify your watch stations or Submarines early. You will impress the hell out of your command and pretty much set yourself up for early advancement, top evals, awards, and stuff like that, but most importantly it HELPS YOUR CREW.