I'm trying to make a roller coaster. I have powered rails, a block at the end, and a button. When you push the button, it makes the carts go, but they leave me behind. I right-click on the cart and it pops me into it, but I'm still able to move and walk through the cart. If I stay still and just push the button, the cart shoots right through me and flies off empty down the track.
You almost certainly have the button too low. Raise the location of the button up a block or two, and the click of the button won't hit the cart instead (currently the cause of your problem it seems).
You might encounter a problem of mine: Right clicking the button while inside the cart seems to sometimes trigger exiting the cart as well. Solution: Left click the button (without a sword in your hands) while inside the cart.