Downtown (Centro):

Plaza de Armas or Plaza Mayor, is the main square of downtown Lima. In the Spanish tradition, it's bordered in four sides by Government/Presidential Palace or Palacio de Gobierno, the Cathedral and Archbishop's Palace, the Municipalidad or City Hall, and assorted businesses.
Museo de Arte- Has permanent and temporary exhibits of Peruvian art, covering pre-Hispanic, colonial, republican and Modern art.
La Iglesia de San Francisco- Fans of colonial churches will be pleased with the architecture of this well-known church in downtown Lima. However, the more compelling reasons to visit this cathedral include underground catacombs featuring over 25,000 real human skeletons (buried between the late 16th and early 19th centuries) arranged in artistic patterns, and also a beautiful library with mysterious, oversized books and spiral staircases in Harry Potteresque form.
Cerro San Cristóbal- This hill is situated close to the Rimac river at 500m above sea level and from here the visitor can contemplate an incomparable landscape: The city with its high buildings, the course of the river Rimac, as well as asphalt roads of the sea resorts and districts. An observation point with an illuminated cross, a site museum and a restaurant that has 'anticuchos' (kebabs) and 'picarones' (fritters), are its best attractions and a good motive for a visit. Entry to the museum costs: Adults: 1 sol Children: 50 centimos. It's not so easy to get here, but many tour companies include a visit here on their itineraries and taxi drivers will know how to go.
Instituto Riva Agüero, in Jirón Camaná, holds two museums: an interesting collection of popular arts, including masks and costumes from current festivities in the Andean region; and a small but nice collection of archaeological artifacts from Lima sites. The Instituto is housed on a very old colonial townhouse, one of the few of its kind open to the public.

Miraflores:

Huaca Pucllana dates back to 200-700 AD. Located in Miraflores and made from small handmade, sun-dried brick, this huge construction is 22 meters high. It is divided into two areas, for ceremonial and administrative ends. Guided tours include a visit through the museum, the small park with native flora and fauna and the craftsmen's house where handicrafts are on exhibit and for sale. Admission is 5 soles for adults. There is also an eponymous restaurant on the site itself, which, though expensive, offers stunning views of the complex, particularly at night, when it's floodlit.
Luis Miro Quesada Garland hall, is commonly known as the Galeria de la Municipalidad de Miraflores (Miraflores Municipal Gallery). Located in the Municipality building, opposite Avenida Larco and some fifty meters from the Parque Central, it is one of the most popular art galleries in Lima. The exhibition rooms are spacious, and the exhibitions themselves are generally very good. Work on show is mainly by local artists, with the occasional foreign contribution. Artists who display their work here usually go on to acquire a good reputation in the national art scene.
Larcomar. It's like a mall, where you can eat, shop, go to the cinema, it has two nice discotecs for the night. The discotecs are open Thursdays, Fridays and Saturdays, and in summer (January and February) also Tuesdays and Wednesdays. Larcomar's principal attraction is its location. It is in Miraflores, in a high hill where you can see the beach while eating or just walking. Best to visit during the sunset.

More:

Museo de la Nacion, along Av. Javier Prado
Museo Larco Herrera, showcases remarkable chronological galleries providing an excellent overview on 3000 years of development of Peruvian pre-Columbian history. Located in a unique vice-royal mansion of the 18th century built over a 7th century pre-Columbian pyramid, is surrounded by beautiful gardens, in Pueblo Libre borough. Popular thanks to its ample collection of erotic pottery from the Mochica culture and the Gold & Jewelry of Ancient Peru exhibition.
National Museum of Archaeology and History, also in the Pueblo Libre borough, quite good and very didactic; the area surrounding it is worth a walk.
Museo de Oro del Perú, Alonso de Molina 1100, Monterrico. Overrated. Peru's gold museum contains a huge collection of gaudy, gold works. Not the best place to learn about Peruvian precolumbian cultures. A large part of a Peruvian 20,000-piece collection of gold is fake, according to the Peru's consumer protection agency. Ref:BBC News Article. A new "branch office" has opened in LarcoMar in Miraflores that is much more focussed and beautifully presented; best of all, of course, is that it is located in the main tourist area.
Pachacamac (site in Spanish) about 25km south of Lima is huge archaeological site containing pre-inka and inka temples (entrance fee 6 Soles). This was one of the major pilgrimage sites in South America, in the years 500 to 1500. The temples are not very well preserved. There are three ramp pyramids and a temple of the sun (the only temple which may be accessed). Unfortunately, the ruins may only be seen from a road, without a pair of binoculars a very disappointing exercise. The on-site museum is poor. From Miraflores, take a colectivo to 'Puente Primavera'. From there, take a 'San Bartolo' colectivo to Lurin (2.50 Soles). Tell the conductor that you want to drop off at the ruinas de Pachacamac.