Intra-Schengen flights are usually, to all intents and purposes, like domestic flights, but with a few subtle differences.

Security is similar to the US. You will need to present liquids (sub 100ml, small clear bag) and electronics (laptop, tablet) separately. You normally have to remove belts with metal buckles, but rarely have to remove shoes. You don't have to present ID.

Check-in and security will be the same as everybody else in your terminal (save for different ID requirements). You won't go through exit immigration, and may not have an ID check on boarding.

On arrival, there'll be no immigration. You may have a customs channel to go through, but you will just walk straight through.

I don't know Barajas airport too well these days, but my rule of thumb is 90 minutes for short flights like this, to allow a little contingency.