First, get the pot that you are moving the plant to and add enough soil to the bottom so that when you set the planted pot inside, the soil level is at the level you want it in the new pot. Next, start filling around the inner pot, firming the soil down slightly. Be sure to work around the pot evenly. If you fill one side first, the hole will be lopsided. Then, remove the inner pot. Gently squeeze the sides of the pot to loosen the soil inside. Slip the plant out of the pot and gently insert it into the hole. Gently tamp down the soil around the plant and add any soil needed to level it out. Water it well to remove any air bubbles and to help the root ball loosen up.
That is it. You will have a happy plant in a bigger pot.