In winter - the beach or wild wadi (if you like water and theme parks). For a picnic I would recommend Creek Park. Sundowners at Barasti Bar.
In summer - hit the malls as they are air conditioned, the Mall of the Emirates for some skiiing, or the new dubai mall for ice skating and the aquarium.
In the evening book a dinner cruise with Bateaux on the Creek for something different.
Go for Desert Drive, many tour operater offer that. its package of sand bashing, Sunset view in sand dunes, camel farming, camel rides, food & drink Plus Belly Dance. It will be a wonderfull evening altogether.
You may want to drive to one of the neighboring Emirates. Fujairah if you like beaches and mountains, Sharjah if you like older markets, and AlAin if you like Desert.
You can even drive to Oman to visit Sohar or Muscat.