Planning to move to Jeddah

Hello everyone,
We are planning to move to Jeddah with my husband and our 2 year old daughter. My husband is a pilot and sometimes he will be out of town for days, so I will have to take care my daughter all alone. We are going to live in a compound which will cover some of our first needs but what if I have to go out of the compound? Will it be ok to take a taxi and go alone with my baby at a mall for shopping or go for a walk at the beach?
What a woman permitted and not permitted to do there?
I would appreciate if someone can inform me about the local culture.