I stayed at Hotel Europa, and took the train as described above. The train is cheaper than the metro and the hotel folks felt the train system is easier to navigate than the metro. I can see why after a long trip many would opt for a taxi. But, for me, with four kids in college and a long list of destinations on my bucket list, I am more of a budget traveler. One thing I would wonder is how long a taxi ride can take during certain parts of the day. The trains and metros have the advantage of not having traffic to deal with. In some cities, I seem to be able to get places far faster on public transportation than I would with a cab.
I didn't spend much time in Madrid, I did day trips, and I would agree that compared to much of Spain, there isn't as much history, and its not really that "old". But, just to provide clarification, guidebook maps often have an area of Madrid around the Puerta del sol labeled "Old Town".
In terms of taxi cab drivers, we didn't use a taxi in Madrid, but we did in Toledo a few times and observed them elsewhere. And the drivers I saw seemed very careful to both be and demonstrate honesty. Before the trips in Toledo, we asked how much to. . . and they gave us really good estimates. I usually round to the nearest euro, but they didn't seem to expect a tip.