If you're going into classic comics, I would recommend
The Greatest Batman Stories Ever Told, which is affordable and features a great overview of excellent Batman stories down through the decades, including Detective 500, which is seen by many--including myself--as the single greatest Batman comic ever published, not the least because its lead story, "To Kill a Legend," is arguably the single best Batman tale of all-time. Another recommendation is Len Wein's legendary series
Batman: The Untold Legend where he basically pieces together Batman's history from all the decades of his existence up until the time of writing into a single coherent narrative.
Batman: Tales of the Demon introduces arguably Batman's greatest enemy and his greatest love interest in a single, sweeping storyline that takes Batman out of Gotham City and almost puts him in the shoes of an Indiana Jones-type. Really good stuff and the characters introduced have significant roles in the Batman mythos to this very day, including in the movies and cartoons. If your kids like Green Lantern at all,
Tales of the Green Lantern Corps is another one that you can't go wrong with; it is an epic story and a great introduction to the characters.