Hi linda, Good question. I think it depends on what the final look is that you are after. If you want to retain wood grain, then staining would be in order.

Paint is a very different looking finish.

I'd go slow with this project. Look at many photos. Many paint some parts of the DR set, like the table base and chairs and let the hutch stay plain. OR paint the hutch an antique color and let the rest of the items stay as is. Can you apply stain over a finished (varnished, polyurethane) wood? I don't know there may be products out there today that allow this?
Here are some painted hutches for ideas http://www.houzz.com/paint-hutch Painted chairs http://www.houzz.com/paint-dining-chairs Painted table http://www.houzz.com/paint-dining-table-

Thanks to both of you. I would prefer the wood grain so stain would probably be the choice but I would have to remove the finish first. That's the part that is making me think twice. I'm not sure exactly how to go about that step.

Minwax seems to be a quality product and readily available http://www.minwax.com/wood-products/stains/

Oh yeah, prep is my least favorite part of any project. :)

This product isn't just for kitchen cabinetry:

http://rustoleumtransformations.com/woodrefinishing/