There’s a great restaurant called Bazaar, C/ San Marcos, 35, just south of the Chuenca metro stop (http://www.restaurantbazaar.com/). The food has an international flavor to it; some Spanish, some Asian flavors but all around good. The prices aren’t too bad either, about 6-10€ per plate, and they have great wines (including house wines).
Last recommendation. San Antón (http://www.mercadosananton.com/) and San Miguel (http://www.mercadodesanmiguel.es/) markets are great places to eat some delicious food, but it’s an even better deal to take some to go. If you are staying in an apartment or going on a picnic in Retiro Park (which I highly recommend), pop into either market. Buy a little of this and a little of that and you will be amazed at the quality of food you get for just a few euros. American readers, please remember that one pound equals 453 grams, so 100 grams of cheese is enough for a good snack for a few people or to last a few days. Add a bottle or two of wine from a grocery store (for as low as 1€) and you’re good to go.
To get in and out of the city, there is an express bus from the airport to Cibeles (Banco de Espana metro stop) and Atocha (Atocha metro stop) for 2€ each way. It’s easy and quick, the driver can provide change, and, best of all, you don’t have to transfer in any of the many metro stations. Check out this site for more Information: http://www.esmadrid.com/en/barajas-airport-express
Also, one other tip about Madrid’s metro system: when you look at the system map, the city seems quite spread out and possibly unmanageable. In actuality, the metro stations are closer to each other than one would think (some as little as 3-5 minutes on foot). I recommend that in the first day or two you are in Madrid, don’t buy the 10 ticket metro pass. Figure out where you think you want to go in the city, how much walking you are willing to do, and go from there.