The Dubai metro is very clean and efficient, albeit a rather long ride to the MOE. Cabs are generally cheap in Dubai and during my trip there hailing them throughout the city was very easy and safe. I'd say a cab to the MOE would cost about 90-110 AED since you'd have to get one of the vans for the kids. Dubai, especially the built up areas such as around downtown, MOE and Jumeirah are all very clean and safe to walk/take a cab, etc in.

Depending on how young your kids are and the time you arrive in DXB (I arrived at 9PM so the metro was empty) the metro may be a better and cheaper alternative.

Unfortunately, I have not visited Abu Dhabi so I couldn't help you there.