There are so many variables. it will depend on his job, the command he goes to, and need. none of us can tell you what to expect because the only certainty in the navy is change. Be flexable and roll with the punches! I sm sure he will communicate to you what is going on as it comes up and he will be the best person to tell you.

His 5 year contract began the day he shipped out.  

I don't know if GSEs have C schools.  At a glance, I didn't see any.  Not much variation on gas turbines. If there is one, and he is chosen for it, he would go to that after A school, in most cases.  However, a command can send a sailor to C school if they need them to learn a specific system.

GSE is a sea going rating.  Chances are high he will get orders to a ship straight out of A school.  Ships deploy, and usually for half the year or more.  Would he deploy right away?  Maybe, maybe not.  But do not expect shore duty immediately.

He should get about a week of leave after school.

Active duty starts the day they ship out to bootcamp.  Honestly, there's no way to tell how things will play out, there are so many variables.  Once he's done with school, he'll go to his first duty station.  Sometimes they can take leave in between.  Once he's at his duty station, he will find out what he needs to know.  One thing no one told my husband is duty days.  He despises them.  He's got duty every 12 days and has to go in, do a watch and stay the night on base.  (I usually take the kids and dinner to him those nights, and we get to spend a little time together).  Other than that....  it's all up in the air.  One guy I used to work with was in 25 years and never deployed.  A guy my husband just deployed with just got back from a 7 month deployment in Feb and they left again in May.  He's only been in 2 years, and he'll have 15 months at sea.  

It's a waiting game, then it's a changing game.  He'll know more once he gets orders.  When he gets orders, he can contact his sponsor, who can fill him in on the details of his new command.