Find out if you can add horse care and handling to your lesson routine instead of just riding. Some instructors want you to learn that and some don't want to bother.
For the kids who really show an interest and make me believe that they will someday own a horse, I do teach everything I can A-Z and try to involve the parents too. However, bear in mind that an instructor has to charge for this just the same as riding.
I also have several students taking lessons just for the status of it. Their interest is minimal so I don't go out of my way to teach them much more than very basic grooming, tacking up and then of course riding.
Please, whatever you do; do not run out and buy a horse until you get some experience in all that it entails. The horse is the one who suffers in the end when people do this!