The best way to get to know your new city is to walk all over and learn the neighborhoods. But if you have bad knees like me a quicker way is to take a tour bus.
The Madrid city bus tour isn’t half bad. It costs a lot more than it should but if you plan your day right then you can use it to your benefit. We splurged and bought the 2 day tickets. You can hop on and off anytime during the 2 days. So we also used it as a mode of transportation since we didn’t have our Abono metro passes at this time. We also took the extended routes; Extended route 1 leaves from Prado Museum at 10:05 am and 6:05 pm, Extended route 2 leaves from Plaza de Neptuno at 12:15 am and 4:05 pm. Like most tours you can listen to the history in multiple languages but remember to bring your own headphones because the free ones are rather uncomfortable.
If you are up for walking then look into one of the many walking tours. You can get some leads from the tourist office at Plaza Mayor. Or set off with a map and get lost in the twists and turns of the old neighborhoods. Madrid center is small and the neighborhoods in the nucleus are closer to each other than it may seem on a map.