Hi PCW
Hopefully you will get some replies from those ladies that live in Riyadh, but in the meantime I thought I might be able to offer a little insight about Khobar (I don't believe Riyadh is that different).
I can drive in our compound, which is one of the larger ones in Khobar (Aramco is like a small town, but you are only allowed there if you work for them), but there is nowhere to go, it takes 5 minutes at 10 miles an hour (the fastest speed allowed for safety) to reach the one little grocery store and I live the furthest house away. Most compounds provide a bus service for ladies, with a schedule of buses to different malls or stores around town. We can use private taxis, but most ladies don't like to do this alone.
Our compound has a school and did have a nursery until it closed down due to the increased deportation of illegal staff. School children seem to travel by bus or private taxi, or Dad takes them to school on his way to work.
On our compound most ladies ride bikes; usually a tricycle if you have little children, with either a trailer on the back or a seat in the basket at the back for the little ones. A few have golf carts, but they are expensive for the very small distances traveled.
Some people have a car and driver, but most of the time it just waits unused, because there is nowhere much to go.
It is a very different life to that in Dubai, most of your activities will be on the compound and you may only leave it to go grocery shopping or to visit friends in other compounds. It seems to be quite pleasant for those with small children, a little less so for those with teens and a lot harder for those who do not have children or have grown ones.
I hope this helps a little bit.