There's no train connection with the airport.
Car and bus
Quito Airport is situated in the northern suburbs of the Ecuadorian capital, at about 7 kilometers from the center. The airport is easily accessible by rental car. The Avenida 10 de Agosto runs north of the city center and turns into the Avenida de la Prensa which leads to the airport.
Several busses run from the airport to Quito's center. There's a stop across the street from the terminal's exit. Buses with the destination 'Amazonas' or 'Juan León Mera' go to La Mariscal (where you find many hotels); buses with the sign 'La Marin' go to the old city center.
Car rental Quito Airport
You can find counters of car rental companies in the Arrivals Hall. Three rental companies have a presence at Quito Airport: Avis, Budget and Localiza. 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.
You'll find many taxis outside the Arrivals Hall. Some drivers use fixed rates for various destinations in the city, others use a meter. Make sure to agree with the driver how you will be charged. The rule of thumb is that if the taxi has no meter or the driver doesn't want to turn it on, you need to agree on a clear price upfront.
There's a parking lot outside the terminal that can accommodate about 300 cars.