What is the process for prior service going back into the Army?

You need to start with your recruiter. Many of your questions should be answered by that person.

In the 14th year of my service in the Navy, I was on the temporary disability retired list (TDRL) from 2001 - 2005. I recovered from my medical issues and chose to return to active duty service. To qualify to come back in, I had to go through the same process as when I first enlisted. I was processed back in through a recruiting office, went through the MEPS, taking all the same tests and physical and approved by the doctor "fit to continue my Naval service." I received Orders from the Navy Recruit Training Center for NAVET/OSVET Training. At OSVET training, uniforms reissued; you'll requalify with arms, physical ready test (PRT), and other reentry training, then you will go up for orders. The RTC is responsible for administratively processing veterans through Veterans Orientation, from ALL branches of the military, back onto active duty. I provided a link below.

You go through the exact same process as you did when you first joined. You draw gear for BCT at BCT. After BCT, you go to AIT and draw gear at your AIT. Depending on your previous rank, you will be housed separate from traditional IET soldiers.

Depending on your rank and how long you've been out. I had to go through reception with new troops but when they went to basic I moved to AIT.