You will have to check with the airline you are flying on to determine if it considers a pack 'n play a freely-checked baby item. If the airline doesn't allow non-transportation items for free, you will have to pay to check the pack 'n play.
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Pack 'n Play®, collapsible cribs and other similar items must be checked at the ticket counter and will be charged at the baggage rate for your ticket.
Children's strollers and child safety seats are not counted as part of the standard baggage and therefore can be easily checked for free. For your convenience, these items may be checked at the curbside, the ticket counter or at the gate. Child safety seats may be brought on board the plane in certain circumstances.
Water for babies is allowed in reasonable quantities in carry-on bags. Remove this item from your carry-on bag to be screened separately from the rest of your belongings.
Don't forget the time you'll spend at the airport prior to the flight, and after landing at your destination, as well. A common rule of thumb is to pack one diaper per hour of travel time. It doesn't hurt to pack a few extra ones beyond the ones allotted for flight time.
Early morning flights are usually less bumpy, which means more time letting your little one out of the car seat to explore the cabin (a must for new walkers). If that isn't possible, and you've got a by-the-clock napper, book a flight that coincides with nap time.