Do I have to use data or wifi to watch a movie I rent from iTunes or will it be stored on my device?

It will be downloaded on the device that you rent it on. Some people have posted that they've needed to be online when starting to watch rentals, so if you will be going somewhere without internet you may want to watch a few seconds whilst you still have a connection - after starting to watch a rental you have 24 hours if you are in the US, 48 hours elsewhere, before it expires.