Thank you for staying and explore the capital of Peru! If you are staying overnight, my best suggestion is to book your stay in the area of Miraflores. From here you can just walk to Park of Love and have the ocean views and the romantic quotes in mosaic tiles. Very near, just two blocks away, you can find Larcomar, the shopping mall facing the Pacific Ocean with handcraft shops, variety of restaurants, and entertainment.
Walk to the heart of Miraflores, or Parque Kennedy or “the cats park” where you can stroll around and try several Peruvian street food such as mazamorras, picarones, emolientes, butifarras (Peruvian ham sandwich), among others. About a kilometer away you will find Huaca Pucllana, a pre-Inca archaeological complex with its site museum, pyramid to walk up, an exhibition of endemic Peruvian flora and fauna including the Peruvian dog, llamas, the guinea pig, quinua, Peruvian yellow chili, different kind of potatoes, and much more.
From Miraflores you can either walk (average 2km) or take a cab to Barranco, a wonderful place to allow good time to enjoy its art galleries, coffee shops, ocean views, bars, and main plaza. Barranco also offers the house of Mario Testino, a well know Peruvian photographer with its exhibition in a beautiful Republican mansion; and Pedro de Osma Museum with its private collection of viceroyal art.
For your evening, the place is the Magic Water Park, far from Miraflores or Barranco (around 8 km), thus a cab ride is recommended. The idea is to visit this park after 6pm when it is already dark to observe the light shows in the water fountains. There are shows at 7.15pm, 8.30pm and 9.30pm and it is closed only Mondays.
Those are my places where I suggest to visit on your own. Highly recommended to go with a guide to the Historic Center where there are plenty of monuments, the area is a UNESCO city and preserves several constructions from the colonial times (1535-1821) rich in history, art and beauty architecture. A guide is important to take advantage of the time there with so many places to visit and focus on your main interests. A guide would also be good to have at Larco Museum to discuss and have a general view and information of the artifacts observed.
Others areas to explore in Lima either with guide or without, depending on your travel style: Pachacamac Archaeological Site, Pedro de Osma Museum, Palomino Islands, Demonstration of Peruvian Paso Horses, Amano Museum, MALI Museum, Museum of National History, among others.
There are companies renting bicycles if you enjoy riding. You can explore the area of Miraflores, San Isidro (for Parque El Olivar) and Barranco. There are bike lines for these three districts. Enjoy Lima!