Thought I would share some of the logistics of flying out of the Madrid Barajas Airport. My flight was direct to the US (Atlanta) on Delta, other direct flights to the US would be similar. Intra-Schengen flights though would have the part after security changed. Neary all US flights leave from Terminal 1.
Getting to the airport:
Your Options are express bus, Metro, suburban train, and private car (taxi/Uber/etc). The suburban train (Cercanias) only really goes to T4, you can take an inter-terminal bus, but not really handy unless coming in on a fast train and you ticket includes the local train.
Express buses from the train stations and a few other locations will stop at each terminal.
The Metro is easy from nearly everywhere, will require at least one change, and a fare supplement to use the airport stops. Maybe the drawback is that the station is at T3, requiring about a 10 minute walk to T1.
Check-in is at assigned desk numbers, departure boards will tell you the desk number.
Security has a couple entries at T1, lines can get long. If you have Delta Skypriority or other airline status, there is a separate, much shorter security area. The regular line was backed up with a couple hundred people, the priority line was basically empty.
Once through security, you go through exit passport control into T1, Gates A and B. Direct flights to the US leave out of A gates, they are about a 7-8 minute walk from passport control (lots of walking at Madrid airport).
There are good services/Duty Free/ Restaurants/coffee places in B gates, A gates has very little.
For US flights, Madrid uses an inconvenient system where they do an additional passport screening into a cordoned off area at the gate. Problem is, there are no restrooms or services in that cordoned off area...so do not get in a hurry to sit at the gate. We usually get coffee and other things in B gates, then wait near the gate until maybe 15 or 20 minutes of boarding. To get out of the "corral" to use the restroom requires a bit of discussion with the security people and taking your passport with you.