West Houston has some areas that are quite nice. I moved here with my family in 1988 and we ended up settling in an area called upper Memorial (making Exxon super close). The zip for the area is 77079. I was a teenager at the time and started at Houston Community College, about fifteen minutes from where we lived (the street is called Fleetwood Oaks and the major cross streets are Highway 6 @ Memorial. Bar none, Memorial is THE area for a safe and comfortable living experience. As a teenager that was my opinion and as an adult I still stand by it.
(Now, if you are talking further west than Highway 6, Katy is a very nice area, too.)
What made the place so nice for me as a young woman is it was and still is SAFE! With young ladies to care for that is likely at the top of your list! Yes, this is a neighborhood where you could walk your dog at 2 AM with NO hesitation. You could let your cat out with no fear. You could park in the street and your car would still be there the next morning.
Best of luck with your hunt and the move-moving is SO stressful yes? If I can answer any more questions, please ask! I am not a real estate agent, just a regular person.