The airlines have very strict rules about traveling with pets. To travel in the cabin they must be in an airline approved carrier that will fit under the seat. They do have to be under the seat during take off, landing, turbulence, etc. In addition to the airline fee, all pets must have a health certificate from a vet will will also cost $$.
This is an excellent website:
Airline Rules Pets Regulations Cabin Cargo Airplane travel for your pet.