I recommend one of two different approaches. Look for rocket kits at your local hobby center. Some are also sold as Target and some at Walmart. You will also need rocket motors, and a launch controller. If you come away empty handed, you can order many things on line at stores like Apogee Rockets. There are other vendors as well. Use your Google skills to sniff them out.
The second method is find a local club and attend one of their launches. They have a launch controller. They will be very pleased to show you what they do and answer your questions. Then build that rocket and come to another launch.
For the little rockets with "A" and "B" impulse motors and empty school playground or soccer field will do nicely. Set up in the center of the largest area you can find. Be mindful of the wind and stay well away from power lines. If your rocket lands on power lines or the school roof, you have lost it. Look here for more on finding a launch field.
It is a fine idea to start building rockets with your child as early as four years of age. At that point you will be doing all the work. But you can, depending on your child's temperament, involve them in launching. My experience has been that they are often ready to start building their own rockets in the fifth grade. This pretty certain for girls. Boys may take a bit longer.
Joining NAR or TRA is a bit expensive but there are many benefits. The primary benefit is exposure to resources, like Sport Rocketry Magazine and organized clubs. Both provide members with insurance in case of accidental damage in the hobby. The National Association or Rocketry (NAR) is focused on Sport Rocketry and contest rocketry. The Tripoli Rocketry Association (TRA) is more interested in High Power Rocketry (HPR). You can do either or both of these with either organization. An NAR section is more likely to accommodate family and youth rocketry experiences. A TRA Prefecture is more likely to have launch equipment and a flying field better suited to HPR.
If you think this is a good idea then I wholeheartedly agree. Step one is to join one of the national associations. Both the NAR and TRA have easy application processes for a new section. They do require that you need other members, so you are going to need to locate or recruit additional members to qualify as a Section (NAR), or Prefecture (TRA).
Starting a Tripoli Prefecture is somewhat daunting. Your Prefect (President) must be Level 2 certified. This is because the TRA is focused on High Power Rocketry. You can only get certification from certified members of one of the organisations. Interestingly, the certification standards for the NAR and TRA are so similar that both organizations honor certifications awarded by either.
Both NAR and TRA provide programs for gaining Certifications defined as L1, L2, and L3. This is all about High Power Motors. You do not need a Certification to launch rockets with 1/2 A through F motors. There are some finer points to this. Many G motors do not require L1 cert. but some do. Also, if you are going to cluster a bunch of motors into a single rocket, you cannot exceed the total impulse defined for a certification requirement. Rocket motor manufactures have a list of certified members and will not sell you a HPR motor unless you are certified at the appropriate level. Some will sell you the motor you intend to use to certify if you give them your member ID. Most of us members will gift you that H or I motor for L1. Some will even share a J, but that is pushing it, because they cost in the $60 range. You can find out all about certification requirements at the NAR or TRA web sites.