We visited the Al Ain Oasis after our morning visit to the Camel Market (just a short drive away), and it was a lush, green, cool sanctuary. There were no trails to walk, but we parked the car right near the entrance (I've placed the note at that spot) and then walked the long empty roads through the oasis. Lots of birdsong and the earthy tropical fragrance of date palms in the air. Some farmers who own the individual plots were busy clearing branches from their trees. We found a shaded bench and enjoyed our picnic lunch before driving back to Dubai.
Some other good information can be found here, like the fact that it's about 3,000 years old.