Learning how people get around in Madrid was a huge shock for me! In Seattle (where I'm from), almost everyone uses a car to get around. There's some public transportation (like buses and a small Metro system), but it's usually unreliable and not very nice to ride in. So, I feel very lucky to use Madrid's amazing public transportation!
Madrid is a very large, dense city where most people live in pisos (apartments), and most young people have roommates. This means that parking can be hard to find and expensive, so most people don't own a car. Madrid's layout can also make it hard to drive because there are a lot of narrow, winding, one-way streets.
Because of this, public transportation is very well-funded, efficient, clean and nice. Most people take the Metro or the bus to get to work, and some people take the regional train (which is kind of like Amtrak in the U.S.). Buying a ticket is very easy--for the Metro and train, you can do so at a vending machine, and for the bus, you can pay the driver. For all public transportation, you can also choose to use your abono (transit card). The abono system is one of my favorite things about Madrid.