It's not clear how old or big they are. While I can see very young kids enjoying shooting with Dad, it's really a lot more practical for them to build usable skills when the gun fits them properly. A gun that works well and doesn't have a cheap, plastic, crunchy trigger is also going to help them succeed. You probably know by now I'm also going to advocate for lead safety by writing that it would be best to shoot lead-free primers and lead-free or at least lead-safe bullets.
I've shot with 10 year old boys that found a Marlin 795 too heavy to shoot offhand. The Crickett can be shot offhand earlier, but it has limited appeal as kids grow older and bigger. Since I did not intend to take them big-game hunting anytime soon, I resigned to bench rifles on a bipod. The good thing about bench rifles is you can get excellent quality rifles made for the adult market and all you need are a low-recoil chambering (giving consideration to the rifle's weight, which might be over 10 pounds with an optic), and an adjustable stock or a stock that can be cut down to the right length of pull.
Really little kids will probably just shoot with the stock under their arm. But if the kid is over 4', the stock can be adjusted or cut down to around 12" and it has a chance of fitting them. I cut down a Marlin and added a bipod. I've had kids use it with the irons, a red dot and a low-power scope. We shoot seated and off a folding plastic Lifetime table. Smaller kids might do better shooting prone. My next step is going to be a centerfire rifle with adjustable Magpul furniture. I was going to go with a Ruger Precision Rifle in .223 which are on clearance right now, but decided in the long-term I would rather have it in 6mm (.243) Creedmore. With an optic, it's a substantial expense, but I think it would be an outstanding rifle for a child that is adolescent age and one they will never outgrow. In the mean time, I want to get a Ruger MPR or another AR15. The AR is a good value right now, easily adjusts for short length of pull, is low-recoiling, and can easily be setup with a bipod and optic for bench shooting. As kids grow, they'll be able to take off the bipod and heavy optic and shoot it offhand. A quality AR might evolve over time, but it won't really ever be outgrown.
I'll be loading my own Starline brass with Fiocchi lead-free primers and Barnes lead-free bullets. In .223 and 6mm, they offer the Varmint Grenades at $0.20 and $0.23 per bullet. The advantage of the 6mm is it can be loaded with heavier, VLD bullets for long range but those aren't cheap to shoot. Now I've been paying $0.04 to $0.05 per shot with .22LR, but even with a semi-auto and a revolver, two kids are only shooting about 100 rounds in an hour-long range session and that's a good day for them. I might take them shooting a few times a month, because they've got a lot of other activities and things to do. Shooting a hundred rounds of centerfire ammo at $25 to $30 a day is a lot more expensive than shooting $5 of rimfire ammo, but it's still not very expensive, and that's for two people. Buying the absolute cheapest ammo you can find might not be the best choice for children. Paying a little more for safety could be well worth it. For any kid to get to shoot even 50 rounds a month is a wonderful priveledge, especially if you take the time to help them make those shots count.
Children can also learn to shoot handguns. Small and medium frame single-action revolvers with small grips can work very well for mature kids that will act soberly with the short sight radius. Examples are the Ruger Bearcat, and Single-Six (etc.). You can also try the Ruger Mark IV to see if their hand fits the grip. The blued versions have an aluminum frame that is lighter. A blued standard model is much lighter than any of the stainless steel versions. Because I'm wanting to get away from rimfire, I want to get a Single-Seven in .327 magnum and load it with .32 Long. There's also a Blackhawk in .32 H&R that can shoot the Longs.