An easy way to see the city is to take the Madrid Vision buses. You can hop on and hop off as many times you want during the day. They run 3 routes: Historical Madrid, Modern Madrid, and Monumental Madrid. You get some explanations about the things you see during the trip, and this in 8 languages.
We found this an easy way to see the city. We bought tickets for 2 days and also used the buses as a way of transportation. One of the routes stopped very close to our hotel. During the summer, the buses run until midnight. We loved to see Madrid by night in this way.