ALL DATES ARE 2015. They have probably changed things a little bit since then, but hopefully this helps someone.
I had a referral from a current Delta employee, with over a decade of service before I applied.
4/10 - Applied for AMT - avionics tech spot in Atlanta which opened this date.
4/17 - Avionics tech position closed at the end of the day.
4/29 - I received an email from a recruiter inviting me in to come in for testing on May 5th.
5/5 - Testing completed at the Delta General Offices along with eight or ten other candidates for the same or similar positions. Spatial reasoning, mechanical aptitude, troubleshooting, and personality (the first three tests are timed - the personality test was not) were the four tests I took. The recruiter told us before we left that all four tests were passed by everyone in our group.
5/11 - I received an email notifying me of two different days I could arrive at the Delta Air Lines Technical Operations Center (TOC) for an interview. I selected May 15th.
5/15 - Interview with a hiring manager and an avionics tech/instructor/project manager at the TOC. No terribly technical questions during the interview - mostly STAR-format team/experience questions.
5/18 - Job offer extended and accepted via phone. Dropped off all documentation and completed the drug test same day.
6/4 - Received a call to select my start date (I gave my employer at the time the standard two-weeks notice). I selected 6/22.
6/15 - My employee number came in the mail.
6/22 - First day of indoc and training.
Just my 2¢: If you have something in your past you are worried about, be honest! I know several candidates who upon receiving their conditional job offer, encountered major issues due to something in their past that they didn't disclose. Some of these people got hired on months later after straightening things out, and others never got hired because of it.
Be on time, and dress professionally during your interview (I went casual business for all my visits to Delta before getting hired, with the exception of my interview which I wore a suit that day).