From the airport, bus 7 (R$3.85) runs to town and on to the bus station. A taxi costs around R$40. Buses back to the airport run along Av Isaac Póvoas and Rua Barão de Melgaço.
From inside the bus terminal, you can get a Centro bus to Praça Alencastro (R$3.85). More frequent buses marked ‘Centro’ leave from outside the bus station and drop you along Av Isaac Póvoas. A taxi from inside the bus station costs around R$35.
Referência and Localiza Hertz are a couple of reliable car rental companies with offices at the airport. The best car for the Pantanal is one with high clearance, though any car can make it during dry season. In the wet season (November to March) you’ll need a 4x4 and off-road driving experience, and some sections may be impassable anyway.