As far as the Begin Center, that’s actually cool not because of the center itself but because of the Bayis Rishon-era catacombs behind it. One of the coolest places I’ve ever been.
That’s a really wide age range, but in case any of this helps- (like, I don’t think your 1 year old is gonna like any of these, but here goes)
Nachal Chalilim hike (very close to Yerushalayim and even accessible via bus, lots of fun, ends up in a cool cave)
Ir David/water tunnels- just make sure your kids aren’t afraid of the dark
Genesis Land- haven’t been there in years, but when my family went ages 5-12 enjoyed it
Ein Gedi- little kids can stay by the lower pools, older kids can go higher
Gamla- the most beautiful place in Israel (and apparently Netanyahu agrees, if that helps) and the little kids can stay in the nature reserve while the older kids hike to the Bayis Sheni-era city. One of my favorite places in Israel- warning, it’s in the Golan, which is also one of my favorite places in Israel incidentally
Water sports in Teveria
Biblical Zoo- really just a nice zoo, but it has pesukim relating to the animals which is a nice tie in
Masada- cool, but you will probably fry
Hula Swamp- nice nature reserve with different kids of bikes for rent
Temple Mount Sifting Project- 10NIS a person and very very cool, sifting through dirt from Har HaBayis (check with your LOR, I guess) and helping the archaeological effort while finding things ranging from bones from korbanos to Turkish cannonballs.
The Talmudic Village in Katzrin (also in the Golan) is quite cool as well, though haven’t been in a while.
Wander the Old City, including the “Fifth Quarter,” the roofs. Lots of fun, just be wary.