I bought a BSA 'Hornet' (I think) for my 2 youngest (10 & 11) last year in .22 and they have loved it. One of them finds it hard to cock but no doubt it will get easier as they grow. Came with a natty little scope (RRP: £35) that made me realize how good my Swarovski is Definitely not a case of 'spending the same on the scope as the gun'.... Great entry level/price gun that will last for years.
Bags of fun and not too expensive and they can both shoulder it / enjoy it with relative ease.
Grab some of those Firebird targets that either explode or create vast clouds of smoke for added fun.
I'd go with a Weihrauch HW25 Carbine in .177 with a simple 4x or 6x fixed air gun scope.
It's a nice quality but inexpensive rifle of the right size / weight with a good trigger, plus it's accurate - nothing is more off-putting to a beginner, particularly a child, than a gun that's horrible to use or is inaccurate.
If you want to push the boat out and buy a pre-charged air rifle, I'd suggest a secondhand Air Arms S200 Mk1, which is what many air gun clubs use for beginners and younger members.
Having been here with both of my sons, I'd also recommend a few reactive targets that can be knocked down and re-set by a pull-string, some spinner targets and a few homemade ones such as water-filled balloons, eggs etc., as these definitely hold a youngster's interest longer than punching holes in paper.
I bought the Weihrauch HW30S for my kids last year, comparatively expensive, but a quality rifle that I hope will be handed down through the generations. The reduced ~8ft/lb may mean will need another once they reach their teens and are introduced to live quarry, but in the meantime makes cocking so much easier for their small arms.