When we travel and AirB&B it, I take a stripped down "kitchen" with us. It includes:
A cast iron frying pan
A non-stick saute pan
A corkscrew
Spatulas (one metal, one nonstick)
A saucepan
An 8x8 baking pan
A 9x12 baking pan
Dutch Oven
I find if I have these things I'm covered for most cooking that I would want to do. Good luck with the move!