Congratuations on your pending move! I may be a bit biased, but I grew up in Katy (West of Houston) and think it is a wonderful place to raise children! Here is some information about Katy schools. Best of Luck!
Inner Houston schools are kind of iffy. With the exception of say the Museum district near Rice and around the Memorial Drive area. If you go to the suburbs such as Katy (I too graduated from Katy) you get a nicer quality of life. My experience is that depending on what you can afford there is quite a few nice areas to look in. Besides Katy look around Lake Jackson, areas near Conroe/Cut and Shoot/ or if you want to be closer Jersey Village or Humble. As you can see I am kind of pointing you to the outskirts of the city. You will stil have amenities associated with the bigger city but you get a more of a friendly feel and better school districts.
Jersey Village is affordable but older neighborhood. The schools rate a little low but it is in a convenient area.
Houston is full of toll roads. Totally expect to spend a lot of commuter time waiting to drive from point a to point b.