Four screws on each vertical stile top to bottom with a decent thread pattern should hold as long as you don't over tighten them. Sounds like the rack won't carry that much weight.
You might see how far down from the top the horizontal inner frame comes and see if you could extend the top of the sides up to allow you to screw into solid wood at the top of the door if you can.
Glue could work, but it's more 'permanent' if you ever want to remove the rack you built.
I would suggest ply or even solid wood rather than MDF to give you better rigidity and screw / nail holding ability if you're using 1/2" material.
To make it easy, try gang cutting the sides by laying them back to back and routing 1/8" mortises across both sides at once for the shelves. that will give you insets for the shelves. Glue and brad nail from the sides into the shelves.
For the fronts, consider cutting 1/4" solid wood rather than the ply. Will sand better and is less likely to splinter from taking items in and out of the shelves.
Here's a quick drawing to show what I was suggesting...
[attachimg#]
And a close-up
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Check out ikea or container store for ideas and inspiration, or even looking on
www.Houzz.com for ideas in their pantry section.
neil