Sheikh Zayed's Private Majlis
One of the interesting things to do in Al Ain is visit the Al Ain Palace Museum, where Sheikh Zayed was born and worked as the Ruler's Representative of the Eastern Region of Abu Dhabi before becoming Ruler of Abu Dhabi in 1966. The palace has been restored with lots of old furniture and artifacts like dellas, swords, khanjars, and cooking pots. It is probably not completely to its original condition (e.g. I'm betting that the air conditioners are not original). But it still is very interesting (and best of all free). They even have some Emirati clothing that visitors can try on (even wooden burqas for the ladies).