ACZoom

The Mexico City metro is by far the most efficient metro of any city I've visited - better than ones in New York and London.
I was quite surprised to see the trains go by nearly every minute at the downtown metro stations, and even in the rush hour, I did not find it too crowded for a non-local to navigate. There is major crowd control, and one-way sections on the platform, as well as reserved carriages for women and children, which can sometimes get confusing because the signs are not always clear, but there will be a policeman to set you straight :-)

The metro is marked with the logo that looks like "m". From the Airport, if you want to take the Metro, walk to the other end of the long terminal, towards Hall A. Get out to the street, and keep walking the same direction, and you will soon come to a Metro sign, the station is Terminal Aerea. Note that there is another station called Aeropuerto, don't alight there when coming back! It will take around 15--20 minutes to walk from Hall F (international arrivals) to the metro station.

The tickets are extremely cheap - N$2 per ride. Buy a bunch, I usually budget 3 per day of stay.
Most people stay at the Zona Rosa or Zocalo, to go there, take the line 5 towards Pantitlan, and then Line 1, which goes to the Zona Rosa. You have to change to Line 2 from Line 1 to get to the Zocalo. There is a lot of information on the metro on the web.

Note that during rush hour, it will be impossible to take any luggage on the metro. Even off rush hours, it is best to use the metro only if you have small or carry-on baggage only.

Lines 1 and 2 are the most used lines. The only station that was crowded during rush hour was Pino Suarez, it connects both lines 1 and 2. Even there, I let a few trains go, and then was easily able to board the third train - waiting less than five minutes! - at rush hour, 9AM. Words to know at the metro: taquilla (tickets counter), linea (line), correspondencia (transfer to another line), salida (exit), andenes (platform), Solo Mujeres (women only), dirección (train direction - last stop).

As for money, there are many ATM machines and cambios - exchange bureaus - to get Mexican Pesos - the nuevo peso, N$, or just $ in Mexico. Bank of America customers can use the Santander Serfin bank ATMs for low fees - there is one in Hall G (international departures) at the airport, and many in the city, for example at Bolivar 44, in Centro Historico, Uruguay cross street.
Exchange US$ if you need, but ATM or Credit Card is best way to avoid fees. Low fee ATMs, or credit cards with even the 3% surcharges they all add nowadays, are the best bet when traveling abroad.

While the metro is just fine even for a tourist, the buses are too complicated, so I did not travel on any bus.