There's no train connection with Montevideo Airport.
Car and bus
Montevideo Airport is situated 15 kilometers east of the city center. The airport is easily accessible by rental car: the Avenida Italia leads directly to the airport. The way is clearly signposted.
Several regular bus services run from the airport to the popular city beach Playa de Los Pocitos and the old city (Ciudad Vieja). Most services have their terminus at the Baltasar Brum bus station.
Car rental Montevideo Airport
You can find counters for car rental companies in the Arrivals Hall. Four rental companies are present at Montevideo Airport: Avis, Budget, Hertz and Europcar. 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.
Officially there's only one taxi company at Montevideo Airport, Taxi Aeropuerto de Carrasco (Tel. +598(0)2 604 0323). You can book a taxi at a counter near the baggage claim area, but you can also do so online. Check the website for (fixed) rates to various destinations in and around Montevideo.
During the construction of the new terminal a new parking lot was also built offering places for more than 1200 cars. It's located right outside the terminal building.