The Army has several enlistment options. All of them are not always available as they are cancel/reinstated as necessary for the Army to fill its enlistment needs.
I am not sure if the First Duty Assignment option is currently available for new enlistees, but for the sake of the thread, lets assume it is.
If that option is available, it is rather easy to chose your first duty assignment. So if you want to go to 10th Mountain in New York, that’s where you will go after initial entry training. However, you will likely be leaving there in 3 years. You only get to chose where you go first. After that, you are needs of the Army (unless, of course, you get to reenlist with the option of assignment choice or duty satabilization, etc).
Assuming you have enlisted and chose the 82nd Airborne division. When you get there, they are going to send you to Airborne school right away. If you fail Airborne school, you will still be in the 82nd, but you’re not going to be very popular there. You’re not going to get promoted, and you will probably be given the absolute worst job or tasking available. You will ride out your time there as a complete turd, until the Army finally sends you somewhere else.
Depending on why you failed Airborne school, you may be given the chance to go back. But otherwise, you are just going to be looked at as a horrible Soldier and a burden to the unit. Then when you finally move to another unit, you will have the oppurtunity to reinvent yourself. The fact that you didn’t pass Airborne school isn’t going to matter much to some Stryker unit, for instance.
If you signed up for the 82nd, but then fail or quit Airborne school, you don’t get to just leave the Army. It’s the same if you sign up for Ranger training or Special Forces training. If you quit or fail anywhere along the way, you just get reassigned according to the needs of the Army.
You always get to pick your “job description” though. If you want to be a cook, you will enlist as a cook. If you want to be infantry, you will be infantry. That is one thing the Army always offers is a gaurunteed job. If you want to be infantry, you will enlist to be infantry and you will be sent to infantry training. The Army will not pull a “surprise, you’re a cook!!” on you.
If you can’t pass infantry training, there are two options. The Army can either send you home, as in “You failed, go home. You’re not in the Army anymore. Try again in 6 months if you want”. Or, they can choose to send you to another job if that’s what you want and that’s what your drill sergeants think is best for you and the Army. I’ve only seen that happen once though. There was a dude who was pretty motivated and passed the Basic Training portion of his initial entry training, but just couldn’t handle the infantry stuff… so we reclassed him and sent him to go be a mechanic. Usually we just fail them and send them home.