Graduating from "A" School fairly soon. What should I expect in the fleet? What are some good tips for being a successful sailor?

Welcome to the ranks. I started out as an E-1. The biggest thing to realize is most things that you'll start on is for a reason. I had my 90 days in the scullery before they retroactively pinned E-4 on me. You learn by doing and the first thing is keeping things squared away. It's critical for shipboard life. You'll learn habits. I was an O-5 aboard an LCC and would always stop to pick a piece of trash off the deck or clear the way for a sailor with a load. You get into these habits so everyone's life is a bit easier and more importantly, safer. Your positive attitude and willingness to heave to will get noticed. Be visible, not invisible. If you have a spare minute, help your shipmate lift that load into place. Actually make the spare minute to do that. So much of your success and reward will be directly tied to how you do your job and how well you take care of those around you. Enjoy the journey.

Never pass up an opportunity to listen and learn. That is why we have two receiving devices and only one sending device.

Attention to detail, particularly in your rating. Start work on your quals as soon as you can, don't wait. Always be willing to learn, and especially be willing to be wrong, if you are. There's nothing wrong with occasionally being wrong -- only if you don't learn from it.

dont fuck up.

First of all, Congratulations on Graduating from A school.
There are a lot of words of wisdom that other men have stated, but I can add, LISTEN, KEEP YOUR EARS OPEN AND YOUR LIPS TIGHT. Do your best, remember, you are PART OF A TEAM when you are on a United States Military Vessel ( be it a truck, sub, ship, aircraft, etc. ), it takes a TEAM to make it run successfully. Be PROUD of being a team member. Make certain you have your shipmates back ( or their "six" ) and they should have yours! Enjoy the fleet. My tour of the Western Pacific was one that I still remember fondly after *cough* years ago. Always be on time, be respectful of your seniors and treat your subordinates as how you would like to be treated by those senior enlisted/officers. It is a GREAT BIG WORLD out there.....Have fun!
You NEVER know who you may bump into down the road! Godspeed young sailor!

Not a Sailor or a Marine but I was advised to also read the Commandant's Reading List, as well as any Army reading lists I could find. If you have the spare time, the mind needs sharpening too.

Have Fun, I'm Jealous been retired almost 20 years now.

Simple answer is PT like a beast. Figure out what it takes to advance in your MOS and surpass that. Always be competitive.

PS, My "A" School is Aircrew Survival Equiptmentman (PR)

Study study study. Get your quals done as soon as possible. Pull your own weight