Taking the ropes to the top path is faster, but there’s a risk of being bumped off if it gets congested, so don’t be afraid to play it safe and go through the marketplace on lap 1.
There are grass shortcuts inside the palace, at the split path section, and after the marketplace (just before finish line). The shortcut at the split path section skips an item set, but it can be done shroomless by doing a charge jump and tricking off the sign.