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You can easily spend this time roaming Dubai itself. Depending on your passport validity you may be entitled to a transit visa (which is for 72 hours). I would get that, leave my luggage at the luggage counters at the metro station and then go spend the day at Dubai Mall.
If the above does not suit you; Terminal 3 would be the best place to spend your time. There are plenty of shops where you can easily whittle away the hours. There are even mini spas and massage stations, a large space with trees and a bench where you can spend some time.
Food options are plentiful as well. You'll find everything from fast food (McDonald's and the like) to sushi stations, Italian restaurants and bars.
Food establishments are spread throughout the terminals; you'll find the main food courts on either end of the terminal, and cafes and coffee shops scattered throughout. They will welcome your foreign currency, but offer you change only in UAE Dirhams. If you plan on coming back to Dubai that's great, otherwise I would recommend paying using your credit/debit card.
Regarding shopping - I am not sure what you would like to pick up. For example, I know some shoppers like the fact that tobacco products are not taxed heavily (same for liquor and spirits) so if that's your fancy there are plenty of shops catering to that. As I do not smoke or drink; I cannot vouch for deals but during my many travels through Dubai these items are certainly popular.
Electronics are another major draw; but personally I did not find the deals that compelling. They are better deals to be had outside the airport.
However, one thing you should avoid buying is gold/jewellery - these items are marked up and are available cheaper at the world-famous Gold Souq in Dubai.
Except for food establishments, and the odd store (for example, Boots and Le Clos) - all stores at Dubai airports are run by the DDF (Dubai Duty Free); so do not expect to see a lot of branded stores and the like.
If you want to relax, the Dubai International Hotel is available within the terminal. You can play billiards, go for a swim, take a shower, get some exercise or visit the spa - easily killing away your time.
Finally - Terminal 3 and Terminal 1 offer short term luggage storage.