Lima is not a place that abounds with a lot of cultural activities. I have been in and out of Lima almost 30 times over the past couple of years and have never found a great amount of really interesting things to do. However, that being said, I have always found the Museo de Oro (Gold Museum) to be an interesting place to spend a couple of hours.
The beaches in Lima are rather dull and the water cold, but the Larco Mar area along Costa Verde in Miraflores is worth a visit. It has several restaurants and bars and various shops that make for an easy way to spend a couple of hours and is a rather busy place on Friday and Saturday nights. Miraflores and nearby San Isidro districts are the fashionable parts of Lima and exploring outward from the Larco Mar area can be worthwhile if you like to window shop.
The Indian Art Market (in Pueblo Libre) is also worth a visit. The quality of stuff for sale varies widely, but with some looking and bartering you could come away with some nice things made by locals for very reasonable prices. I have been able to find nice silver pieces and alpaca sweaters and the like in several of the stalls. Look carefully and shop around before buying. Very un-touristy place, but language should not be a problem. Also, the Mercado Central (near Ayacucho and Ucayali) is an experince. Keep and eye on your wallet and enjoy the madness. Everything from antiques to can goods for sale.
Plaza de Armas, the central plaza and cathedral, is neat to see but do not venture down there in the evening. Lots of petty theft, etc in the areas immediately off the plaza at night.
Regarding food, Lima has a number of good places to eat. Miraflores and San Isidro districts again have the best places. For some reason I have yet to figure out, Peru has an inordinate number of good Italian restaurants throughout the country. Not just in Lima. Ask the folks at your hotel to recommend a good place nearby. For an especially good meal head to either La Rosa Nautica at the end of the pier in Miraflores (seafood) or El Rincon Guacho (near Larco Mar) for Argentine steaks.