Spanish meal times are later than in the US - lunch is typically served between 2:00 and 3:00pm and dinner anywhere from 8:00 to 10:00pm. Keep this in mind, as many restaurants will not open for lunch before 1/1:30pm and will not open for dinner until around 8:00pm. Lily learned this the hard way her first month of study abroad back in 2012!
Unlike at home, tipping in Europe is neither common nor expected of you. In "nicer" restaurants, we usually leave 5€ at most when its just Pablo and I, and 10€ at most with a larger group. In normal, everyday casual places, just a Euro or 2, or even just rounding the check up to the nearest Euro is more than enough. It is not common to tip in taxis, at hair or nail salons, etc!
Spaniards typically greet and introduce themselves to others with 2 kisses on the cheek instead of a handshake. (We also let the Spaniards know that Americans typically do not do this and might be surprised if they lean in for a smooch on the cheek when meeting and greeting each other)
Madrid is a VERY safe city at all hours of the day, BUT unfortunately petty theft is pretty common especially in more touristy areas and in the metro. Make sure to keep a hand over your purse/backpack and keep the zipper somewhere within eyesight. When at reastaurants and cafes, do not leave your phone or important belongings on the table without keeping a hand on top of them! Be wary of people that come up to you "collecting signatures" - this is a typical scam, and while you're distracted & signing your name, they usually send an accomplice to snatch your purse/phone/whatever they can grab. The best thing you can do if one of these people comes up to you is hold tight to your belongings, ignore them and walk away - they won't chase after you!