Corumbá is just across the river from the Bolivian border town of Puerto Quijarro.
The Bolivian Fronteira Border is served by taxis on the Bolivian side. Typically a taxi from the train station to the Fronteira/Immigration costs about 5 Bs per person and takes around 5 minutes. Be prepared to share your taxi with other travellers. Bolivian Immigration is located on the border from which it is a short 100 yards walk across the border to the Brazilian buses and taxis.
The Brazilian Fronteira is served by taxis, motos and the city bus. A taxi to the center of Corumbá will cost you R$20-25. The bus runs once an hour, -at 45 minutes past the hour from the Fronteira and 15 minutes past the hour to the Fronteira. The cheapest cikets are from the cafeteria next to the bus stop. Double the price on the bus.
Brazilian Immigration is no longer in the bus station, you can catch the bus from the center which is destined for "Fronteira". The immigration is now at the boarder. Be advised that the immigration office on the Brazilian side closes at 1300 on a Sunday and you cannot exit until getting stamped. Queues can be very long.
There are connections by bus to Campo Grande 5 hours, São Paulo 20 hours and Rio de Janeiro 25 hours.
Corumbá has an airport served by TAM (http://www.tam.com.br/). The Bolivian town Puerto Suarez also has an airport.
