Not sure when you are planning on coming but if it is at the beginning of December, then it will be busy as its National Day to celebrate the creation of the UAE so hotels will get full (and expensive).
Abu Dhabi Culture and Heritage lists various activities and sites http://www.adach.ae/en/
Abu Dhabi Tourism Authority http://www.abudhabitourism.ae/en/Default.aspx
Must sees would be:
Free public beach - plus there are some areas that you can pay a small amount and get a sun lounger, towel, umbrella etc. located on the corniche but the sea is hot - its like swimming in soup!
Sheikh Zayed Grand Mosque - fabulous mosque which is a must see and has well organised tours available http://www.abudhabitourism.ae/en/portal/zayed.mosque.aspx
Yas Island to see the F1 circuit although if you are coming this week, its the F1 and therefore tickets are very expensive and probably not available!
Friday Brunch at one of the hotels
Trip out into the desert with shisha and a traditional dinner organised by one of the 4x4 groups which can be arranged via a hotel or a lot of people say Arabian Adventures are good http://www.arabian-adventures.com/
Shisha - try Special Cafe which has three locations on the corniche, or go to a swish shisha tent at one of the hotels such as the Shangri-La or Emirates Palace
And if you have any left over time:
Emirates National Car Museum has some corkers in it http://www.enam.ae/ver4/index.php
Jebel Hafyeet in Al Ain - about two hours away is great - up a mountain, wonderful views, cool breeze!
Shopping - lots of malls
Trip to a camel farm! over 6 years ago