Jumeirah Beach Walk is a short taxi ride away - over 1 mile of shops, cafes and restaurants - only a few are licensed though.
IBN Battuta mall - short cab ride away. Lovely architecture, lots of shops and nice places to eat. Not licensed though.
Mall of the Emirates - where you can go skiing if you want. 10 minutes by cab, lots of nice places to eat, some of which are licensed.
Dubai Mall - this area is known as downtown (although it isn't really downtown and is brand new). Souk al Bahar, Burj Khalifa and the evening fountain show. Over 100 restaurants, many of which (in Souk al Bahar) are licensed - in the mall they will be unlicensed.
The creekside and the souks - about 35 miutes by cab. Great to wander around and explore for the day. Here are two maps
Take a cab to point 1 on the first map where the museum is and then take your pick of the things you want to see. Plenty of lovely palces for lunch in this area. You can catch an abra across the creek for 1 dirham per person to visit the gold and spice souks on the other side
There are so many restaurants to choose from that it is really difficult to recommend any. You will find all of the cusine styles you mention in all of the places I have mentioned - so you maybe best just seeing what you fancy on the day.
The duty free allowance is 4 bottles of anything per person or 4 boxes of beer or a mixture - so 2 bottles of wine, a bottle of vodka and a box of beer per person if you want but perhaps that's quite a lot for 4 of you.
A desert safari one evening, afternoon tea at Burj al Arab hotel, a ride on an abra across the creek, a trip up Burj Khalifa and watching the evening fountain show at Dubai Mall would be on my list of things a first time visitor should try and do.
Can I suggest that you take a look at the Things to do in Dubai section of the top questions, the city guide and also the section on Dubai Marina and Jumeirah Beach Walk.
And also take a look at the section on the Entertainer voucher book to see if it will be worth it for you to purchase a copy of that from any supermarket.