To start out, you have to go through recruit training and OSUT (infantry training) or medic school. Afterwards, when you get into your billet, you would take Air Assault School. There is also Flight school if you're a pilot, engineer's school, Mechanized Infantry, and Weapons (weapons squad) courses for specific MOS's.
Further after that, there are several leadership courses and combat life saver. If you get the required rank and transfer to TACP (usually requires application) then there is a TACP course, which is the "pathfinder" course.
That's about it as far as additional classes. It's a lot to take in and most of them are well worth it.
If you want to advance, show up to all your events, take the various courses, and gain experience. Take initiative and sign up for collateral duty billets (paperwork processors, instructors, etc.) Then finally, if you meet the rank qualification and applications open for something you want, apply; do that even if you don't feel that you're experienced or "ready" enough. At worst, you put your name out there to higher ups, and they like that if it's initiative in putting in an application or other "good" deeds.
Biggest thing is showing up. If you don't show up consistently to trainings and operations, you will get nowhere and will probably be sent to the reserve platoon for a period of time.
A unit such as this really takes dedication if you want to get the full experience. That being said, never EVER put the unit over real life stuff. This is just a game in the end, and we're all here to have fun and then sleep in our own beds and go on with our daily lives.
If you have any questions you can PM me. Hope this all helped.