You're going to be disturbing the roots, so keep that in mind. You will stress the plant and there's no getting around that. Some plants take it better than others.

If you are repotting because it's outgrown it's pot, then you will probably find a solid wall of roots in the pot. Do your best to pull it out without disturbing too much. You will want to unshape the roots so that they grow out into the new soil. Roots which have taken the shape of a pot will continue to grow in that shape and you will receive little benefit from the new space unless you unshape the roots.

Depending on the plant, I sometimes do this very slowly and carefully, or I sometimes rip the roots apart. Hardy, easy-to-root plants don't need as much care.

In my experiences I've found that after repotting, it helps to water. This helps settle the new soil around the roots and makes it easier on the plant. Just don't drown the plant.

Try to keep the plant out of direct sun for a day or so. It's easier on the plant if it gets indirect sunlight to settle in.

You're going to be disturbing the roots, so keep that in mind. You will stress the plant and there's no getting around that. Some plants take it better than others.

If you are repotting because it's outgrown it's pot, then you will probably find a solid wall of roots in the pot. Do your best to pull it out without disturbing too much. You will want to unshape the roots so that they grow out into the new soil. Roots which have taken the shape of a pot will continue to grow in that shape and you will receive little benefit from the new space unless you unshape the roots.

Depending on the plant, I sometimes do this very slowly and carefully, or I sometimes rip the roots apart. Hardy, easy-to-root plants don't need as much care.

It sometimes helps to have the plant be a little under-watered before repotting. The leaves and branches are easier to move without damaging if they're a little sad and floppy. Don't take this too far (or at all with some plants!), but if the plant is plump and stiff with water the leaves can break off more easily. This is especially true of succulents which can handle going dry for a bit before repotting. If you water at the same time every day, plan to repot right before you water.

I will also sometimes water as I am unpotting to help loosen everything away from the pot. If the soil is hard and dry it will tend to stick to the pot.

If possible, lightly squeeze the sides of the pot to help loosen the soil and roots from the sides. If it's not possible, you can try poking something from the bottom to push the soil up and out of the pot.

In my experiences I've found that after repotting, it helps to water. This helps settle the new soil around the roots and makes it easier on the plant. Just don't drown the plant.

Try to keep the plant out of direct sun for a day or so. It's easier on the plant if it gets indirect sunlight to settle in.