There's no train connection with Lima Airport.
Car and bus
The airport is situated 16 kilometers northwest of Lima's city center. The airport is easily accessible by rental car: the Avenida Faucet runs in front of the terminal and you can reach the center of the Peruvian capital via this road.
There's no bus station in or near the terminal. You need to walk to the stop on the Avenida Elmer Faucet to catch a city bus. Minibuses (combis) also only stop on that road, not right in front of the terminal. The combis stop along the main roads in Lima (Avenida Abancay, Avenida Tacna and Avenida Wilson).
Car rental Lima Airport
You can find counters of car rental companies in the Arrivals Hall. Three international rental companies are present at Lima Airport: Budget, Dollar and Hertz. You can also book a rental car in advance using the service provided by EasyTerra; they compare the rates of several providers in a single quote.
Three taxi companies are allowed to transport passengers from the airport: CMV, Mitsoo and Green Taxis (Taxis Verdes). Their counters can be found just outside the exit of the Arrivals Hall. You can book a taxi here, but you need to negotiate the price yourself. Agree on a clear price; sometimes taxi drivers agree on a price while it's not clear if they mean 'dollars' or 'soles' (the Peruvian currency). Also watch out for 'helpful' men wanting to place your baggage in the trunk of the taxi. They charge ridiculous prices for a job that takes a few seconds and the taxi driver will refuse to leave until you pay the tip.
The short-term parking lot is located right opposite the terminal. You can park for free for 10 minutes, after that you'll be charged 4.50 soles per hour. The large long-term parking lot costs 35 soles per day and is carefully watched. Shuttle buses run from this parking lot to the terminal.