Answer: The first thing I would recommend is perhaps looking into a good book on houseplants. I like "Houseplants for Dummies" and "The Houseplant Expert". Both are fabulous sources of information. The first step is to make sure when choosing your new container, you move up only one size at a time. For example, move from a 12" pot to a 14" pot. Also, be absolutely sure the new container has excellent drainage. Examine the roots of the plant. If the roots totally encircle the plant, cut away a half inch to one inch slice of roots with a sharp knife from around the pot shape, and also from the bottom. Then, gently loosen the soil with your fingers. If the roots are not bound around the pot, you can skip all of that.
Here is the quickie description of repotting, taken from "Houseplants for Dummies":
Remove the plant from it's pot.
Pour enough moist potting soil into the base of the new pot to bring the plant up to the desired level.
Slip the old pot into the new one and center it.
Pour in potting mix all around the old pot, compress lightly with your fingers, then remove the old pot. (The old pot leaves a nice mold of the plant's root ball)
Slip the rootball into the mold and smooth the potting mix over.
Water well, allow to drain.
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