Take the official "Sitio 300" sedan taxi to the Centro Historico. If the traffic is bad, get out in front of Bellas Artes and from there you can walk to the Zocalo in about ten minutes. Visit the Templo Mayor (don't miss the onsite museum), cathedral and the presidential palace (to see the Diego Rivera murals, bring ID). If you are in the Bellas Artes area, visit that building too.
Lunch is late in Mexico, but be at the restaurant around 2pm to ensure a seat. There are no really top-notch places to eat in the Centro Historico, but Cafe Tacuba is fun, with a good set menu and musicians passing from table to table. Girasoles next to the art museum has some reasonably inventive food and the Opera Bar has bags of atmosphere.
The Centro Historico is dead at night, but you could hop in a cab to Condesa where there are heaps of chic bars etc. Get dropped off at the ultra-trendy Condesa DF Hotel, which is a good place to drink and people watch. If you want an evening meal, just a couple of blocks away is Rojo Bistro (on Parras) or across the park, Bistro Mosaico (on Michoacan, between Amesterdam and Insurgentes). There is a 'sitio' cab stand across from Condesa DF to get you back to the airport, or get your restaurant to call you one. Since you are a newbie to the DF, I don't recommend hailing a cab in the street, especially at night. However, the Condesa area itself is very safe and there'll be plenty of people out an about, especially on a Sat evening. Have fun.