I grew up going to Major League baseball games, and so have my kids. My dad has a season ticket package that he splits with some other guys. When there is an afternoon game, he likes to take one of the grandkids with him. This has gotten to be a good tradition with them.
But if your family is ready to take in a game, you don't need to be able to purchase season tickets. Throughout the season family nights are offered. You can often find discounts with leading products as well.
I've heard that Minor League games are much cheaper and just as fun, if not more!