Lima is vast and sprawling, but most visitors spend their time in a fairly small triangle — the historic centre, Miraflores, and Barranco — and for that, taxis (or, increasingly, Uber and Cabify) are the simplest option. Metered taxis are rare: with a street cab you negotiate the fare before getting in, and foreigners should expect to be quoted double the local rate. Uber is cheap, honest and widely used.
Lima's one modern Metro line (Linea 1) runs along the eastern edge of the city and is not much use for sightseeing; a second line through the centre is under long-running construction. The cheap and fast Metropolitano bus-rapid-transit system is genuinely useful — its main line runs the full length of the city from Chorrillos in the south through Barranco and Miraflores to the historic centre and beyond. The local micros and combis (minibuses) are chaotic, cheap, and not recommended unless you speak Spanish and have time to work out the routes.
Safety: Lima is much safer than it was twenty years ago, but pickpockets and phone snatchers are still a problem in the centre, at busy markets, and around bus stations. Avoid unmarked taxis at night.