Use a small craft paintbrush to get in all the nooks and crannies. This is extra important if they were hung on a flocked tree. Flocking is pretty to look at, but it may discolor ornaments over time.
Metal hangers can scratch the delicate painting and coatings on ornaments, so best to store the hangers separately.
You can store the ornaments in their original boxes if you have them. I've even been known to buy empty vintage boxes when I see them and stack them as fun Christmas decor.
A cardboard/pulp egg carton makes a great place to store smaller ornaments. You can just use the ones from your local grocery store as long as they are clean and dry.
You can purchase storage boxes with divided compartments, like these canvas ones with acid-free dividers. DO NOT use plastic boxes since the plastic can trap moisture and damage heirloom ornaments.
When you inevitably run out of canvas storage boxes, egg cartons, etc., you can wrap your ornaments in acid-fee tissue paper and carefully stack them in cardboard gift boxes or the photo storage boxes from the craft stores.