Depends on what you are looking for, there are a number of things you can do:
Cultural tour: downtown Lima has some old buildings and cathedrals near the presidential palace. There are also a number of museums that might interest you.
Zoo: by San Miguel there is a pretty big zoo and the ticket is cheap. They have a good range of wild animals to see.
Sports: tennis courts are available in Miraflores. Also you can do some jogging along the coast. If you are into Martial Arts (boxing, submission, etc.), there a number of gyms you can train. I go to one near the end of Jose Pardo of Miraflores.
Adventure: South of Lima by Ica you can ride sand dune buggies. Further down South you can find Nazca lines.
Beach: South of Lima by Taxi for around 40-60 minutes, you can get to some of the beaches. Peruvian beaches are not nice except in Mancora, just to be warned.
Disco: Clubs in Larco Mar (Gotica and Aura) are the local's favorites. Other places are in around Barranco.
Movie theatres: Larco Mar and Jockey Plaza have the best cines. In Peru there is no iMax.
Finally and most importantly - Safety issue:
Neighborhood to avoid all together: San Juan de Miraflores (Not Miraflores), San Juan de Lurigancho, Los Olivos, Puente Piedra, La Victoria (at night), first few blocks of Av. Brasil (at night).
Taxi: Don't trust taxis, especially Taxi Seguros (Black ones in the airport, yellow cabs around Larco Mar). Not only they are more expensive, they are actually MORE dangerous, because criminals know that people take those taxis are usually foreigners and people who have money. Negotiate as hard as you can. Not only it brings down the price, it also gives you the appearance of being poor and hard to deal with, so criminals are less likely to bother you.
If you have specific questions don't hesitate to contact me: [email protected]