Condesa, Roma, Centro Histórico, Coyoacán, and Polanco stand out as Mexico City's hippest neighborhoods. If you're looking to do some barhopping and want to foot it, you can do so in La Condesa. The Zona Rosa has lost ground to Condesa, Roma, and Polanco in the past few years, but it's still packed on Friday and Saturday nights, and everything is within walking distance. Niza, Florencia, Londres, and Hamburgo streets are teeming with bars and discos.
Night is the key word. People generally take in dinner and a show at 9 or 10 pm, head to bars or nightclubs at midnight, then find a spot for a nightcap or tacos somewhere around 3 am. (Cantinas are the exception; people start hitting them in the late afternoon and most close by 11 pm.)
You should have no trouble getting around on your own Always take official hotel taxis, sitio (stationed) taxis, or use the safe taxi apps Yaxi or Uber; it can be expensive to barhop this way, but your safety is worth the cost.