The center of town has a lot of action during the day and a little bit of action during the night. It features Congresso (congress building), Plaza de Mayo, Casa Rosada (like the white house in the states), the Metropolitan Cathedral, the Oblisk (located on 9 de Julio), and the world famous Teatro Colon. You can walk up and down Avenida de Mayo from Plaza Congresso to the Casa Rosada (or vice versa) in about 25 minutes without stopping. While you are walking from Congresso to Plaza de Mayo you will cross over the world's widest street, 9 de Julio. By the time you go half way across the street the light changes, and you have to wait about 3 minutes for the light to change to be able to cross the 2nd half of the street.
Subte (Metro)
The supte runs up and down Avenida de Mayo. You are able to hop on the subte right by Congresso and take it all the way to Plaza de Mayo. There are four stops in between these two destinations that allow you to get off and see different locations. If you have time I would just suggest to walk the streets, because there is a lot of stuff you can see in such a short amount of time.
Food
There are a lot of cafes and restaurants that are located up and down 
Avenida de Mayo. Most of them have seating outside during the day and into the early evening. Most of the restaurants will close down in this barrio at about 7pm, because a lot of the their business is from the workers who commute to the center of town. There is a mom and pop sandwhich shop that I went to about every other day that is located on Avenida de Mayo and Uruguay (in between Avenida de Mayo and Rivadavia). You can't miss it because there is always a line coming out of the store during lunch time. They have all different types of sandwiches, but I think the best sanwhich is their milenesa completo! A sandwhich here is only 3.40 pesos (just over $1 Us Dollar). There are also plenty of pizza parlors in El Centro that sell empanadas and pizza by the slice. Of course, don't forget that almost every restaurant sells steak!
Make sure you check out the Havana store that is located right on Avenida de Mayo and very close 
to the side street Florida. This place is always packed. They sell the traditional Argentine deserts called Alfajores. This desert is so good! Another big seller is called dulce de leche. It basically looks like and tastes like caramel. You can put it on basically everything and it makes it good. They also sell cookies, cakes, coffee, and bread! (Map from allaboutar.com)
There is only one burrito restaurant that I know of in Buenos Aires, and it is called California Burrito Company. It is located off of Florida too. This place was started by a couple of travelers from the States who decided that they wanted to stay in Buenos Aires and work. They do have happy hours and music here during the week! Check out their web site.
Bars/clubs
I only went to one club that was located in El Centro. It was a Brazilian club called Maluco Beleza. I had a great time every time I was there (about 5 times). The best nights to go are on Wednesday and Friday nights! It is located on Sacramiento y Rodriguez Pena. They dance Lambada here, which is a Brazilian style of dance! If you are interested in learning how to dance Lambada, you can show up for an hour class that starts at 10:30 pm on Friday nights. Then you can stay afterwards to practice with other people when the club starts to get packed around 12:30am.
About 2:30am, the upstairs part of the club opens up and plays more contemporary Latin beats. I would recommend not showing up to this club until after 12 am. This club stays open very late so be prepared to have fun! The drinks are pretty cheap too!
Shopping
Very close to the Plaza de Mayo is Calle Florida, the main shopping street for tourists in Buenos
Aires. Most of the shoppers on this street are tourists so the prices are a little bit higher than in other locations (compared to Abasto). There is a huge variety of stores: electronics, cell phone companies, clothing, sports stores, jewelry, etc. Galeria’s Pacifico shopping mall is also located on Florida. It is a very nice mall that has a lot of shops and a food court located at the bottom of the mall.
Calle Florida also has a lot of banks with ATM machines. The HSBC bank that I 
used to with drawl money from is located here. I used this location multiple times to avoid ATM fees (I wrote a section on how to avoid ATM fees in a seperate blog). Please be careful about counting your money on the street or flashing pricey items, because there are a lot of thieves on this street that are looking to rob you.
*Every Thursday at 3:30 in the Plaza de Mayo, the "mothers of the plaza" walk in a circle around the plaza in protest to the governement. They want to get answers from what happened to the missing people during the Argentine Dirty War.
