If your staying in the center of Corumbá, everything is walkable, with the exception of "O Cristo" which is on a hill that overlooks the city. The bus for "El Cristo" will take you close to the steps which lead to the top where you can have a view of the town.
Prepaid bus tickets are freely available from people in or outside the bus stations These people do not work for the bus companies This is the preferred method of buying tickets for the bus. Typically these are a gold colored plastic coins but more recently they are introducing paper tickets. These tickets can also be used in many small corner shops to buy goods just as you would use money. It is advisable to check what their exchange rate is first.
If you need to get out of the center then a common method of transport is the moto taxi. You need to know where you want to go and can go a modest distance for R$1-2.
From the main coach station there are regular buses daily to Campo Grande and São Paulo; these are served mainly by Andorinha (http://www.andorinha.com/).
