Call your local airport and check that you can put additional stuff in the bike box with your bike. Amtrak told me I could not and made me unpack everything in baggage claim (glad I showed up early).

Reusable shipping bags are a great way to transport stuff. They can be thrown out once you reach your destination, and re-purchased in most areas for the return trip.

Cab rides are your friend, in this case, but the occasional warmshowers host close to the airport may be willing to pick you up, too. Cab companies should give you a minivan if you ask for no additional charge.