You can get five and three gallon buckets at hardware stores, Home Depot, Lowe's, etc. Just drill holes it the bottom of the buckets for drainage. You can also get regular plant pots at garden centers. I don't think that it matters much what the pot or bucket is made of as long as its not clear (not good to have light on your roots). Also avoid making a pot out of wood because wood absorbs water and tends to get moldy.
Yes it is good to transplant from small pots into bigger pots. What this does is allow the roots to fill the smaller pot before putting it in a bigger pot. If you put a seedling straight into a 5 gallon pot they don't have the root mass to absorb the water after a watering. The soil stays wet and the roots have a tendency to rot. Hope this helped.