If they charge to check bags, how much is it for the first, and how much for the second? How much is charged for carry-on pets, if that is your only carry-on?
NW allows one checked bag under 50 lbs free of charge.
This excludes any service animals, of course. Your pet is NEVER considered a carry on allowance. As for your pet, only one or 2 pets are permitted in the entire airplane, so you must make a reservation and the fee is $80 one way for NW and that is for under your seat. You must provide a current travel certificate from your vet, including rabies vaccine.
Also, during the hot summer months, some airlines will not take pets at all. Southwest never accepts pets. Just FYI.
http://www.nwa.com/travel/luggage/checked.html#freeluggage
http://www.nwa.com/travel/animals/carryonpets.html
If you are traveling on one of their partner airlines, call NW directly.