Gravity on the cart
FCEng84,
Gravity acts on the drinks cart. The reaction force from the floor of the aircraft, through the wheels of the trolley prevents the trolley from plummeting to its doom. In non accelerating flight, it exactly balances the weight of the cart. It's real and you can feel it, just as gravity acts on you and the sofa equally, but there is a reaction force keeping you from falling to the floor - as you say, this certainly means that you can't tell whether you're sitting on the couch, or sitting in a non-accelerating aircraft.
Gravity also acts on the aircraft: per kilogram in exactly the same amount as on the drinks cart. The lift force from the air flowing over the aircraft's wings prevents the aircraft from plummeting to its doom. When there is no acceleration, it exactly balances the weight of the aircraft (which includes the weight of the cart), while thrust exactly balances drag.
To push the cart forwards up the aisle requires work to be done, because it's being taken uphill. When descending, the cart requires work to be done to push it backwards down the aisle, as again it's being taken uphill.
If the aircraft is accelerating, forward, back, or in a turn, then another force - the mass of the cart multiplied by the acceleration - adds vectorially to the cart's weight. This can make it easier or harder to push depending on its direction. As you, say accelerating in a climb will make the push harder, slowing in a climb would make the pushing easier. Just as it does walking down the aisle of a courtesy bus. If a zero-g free-fall trajectory was being flown, then the cart would float up the aisle by itself after being given a gentle push, as the acceleration of the aircraft is chosen to match that of gravity, and the cart free-falls within the plane.