When I moved into DC, I did the car inspection first – you do not need the DC licence to do this. Also, you can make an appointment online or monitor the camera that shows the lines and head over when the line isn’t that long. Also switch your auto insurance to DC before you go to the DMV – DC law requires proof of valid DC insurance for the car before you register it.
Then I went to the DMV and got the rest done in one shot. I second the bring every possible form of documentation. And bring the ORIGINAL documents, no copies. I think they say you can’t use the same document to prove two things. For example, you can’t use your current driver’s licence to both prove that you can drive and that you are who you say you are. I basically wrote down on a piece of paper what all I needed to prove and how I was going to do it, then double checked before I headed out that I had each document with me.
I was in and out fairly quickly, so it wasn’t that bad. Good luck.