What comes after basic training in the us army?

There are a couple of answers to this.

Most soldiers will go straight to Advanced Individual Training, where they learn the skills needed to do the job they enlisted for.

If you joined an MOS that requires language training, you do that before AIT. The installation you go to is the Presidio of Monterey, California.

Soldiers who joined MOS 18X (Special Forces Enlistment) go to a lot of schools: Infantry School, Airborne School, then Special Operations Preparation Course, Special Forces Assessment and Selection, and finally the Special Forces Qualification Course.