You will need to make a reservation for your pet to fly with you in the cabin, as the airlines limit the number of such pets that can travel on any given flight. There also will be a chargetypically $75-100 each wayfor the pet. Finally, the pet will need to be in an approved carrier that will fit under the seat in front of you.
I'm also pretty sure that traveling with a pet requires that you check in with an actual agent at the airport and to pay the pet fee at that time; but I'm not sure where that leaves you with regard to online check in if that's important (like with Southwest). It's possible that you still could check in online for yourselves (to get an A pass on SW or for some other reason) and then check in for your pet at the airport. It's also possible, however, that having a pet reservation linked to yours will lock you out of the online check-in capability.
All carriers have slightly different rules for the service, so you should check with your airline ASAP to find out more and to make arrangements.