We have two types of cards that allow us to travel in Madrid and elsewhere in Spain at discount rates. The Abono Tercera Edad Transportes ("tercera edad" starts at age 65) is an ID card that gives us unlimited transportation throughout Madrid via bus or metro (subway) for 11 euros per person per month. Because each round-trip bus ride is usually 3 euros per person, and we take 12-20 bus or metro rides per month (36-60 euros), this is a good savings. One day, we took an 8-euro bus ride to Chichón, with nothing more than our Abono.
The other travel card is the Tarjeta Durada, which allows us to travel on Renfe trains outside of Madrid for a 20-40% discount.
With these two cards, we can go just about wherever we want without spending much money.