The brazilian airline TAM connects Campo Grande with the main airports of Brazil.
TAM is a very nice and comfortable airline with pretty good legspace and good inflight service.
If you plan to travel around Brazil they also have quite good deals on air passes.
They have also recently become members of the star alliance so you can collect points aswell if you are a star alliance member.
They look simple but are safe,the guides are experts and you need one otherwise just to look without understand,it's a pitty,because a guide can explain the inundations influence in any point,and take you straight to the crocodiles points( don't worry...)!!
To get into the area of Pantanal by car,use the BR-070,between Cuiaba and Rondonia (BR is the simbol in Brazil for a highway,and has always a nbr included ).
To get into a trip to the National Park,you'll need to take a boat in special guided tours,departing from Jofre Port or using the Transpantanal boat sistem.
This is the only way to get to Pantanal, by truck like this...
Can you imagine riding for 5 hours on no roads?