1. Go lightweight if you want it for kids.
2. Keep your factory barrel, and use it. The heavy bull barrels are a waste of money. I have bulls in steel, stainless steel, and carbon from Volquartsen, Clark, Clerke, GM, Lilja, and have had others. They don't do very much that my rack full of factory barrels can't do. Some of them are more accurate than the factory barrels, some are not. And even so, the type of use I put a 10/22 to does not make the fat, heavy barrels worth it. Silhouette is different and I do have a very accurate, heavy barrel on my comp. gun. The others get factory tubes worked over somewhat at times, but still factory barrels. The expensive barrels are sitting in the rack.
3. For a stock, go lightweight and avoid an offhand specific design if the gun is to be a multipurpose tool. If you are going to build 3-6 10/22s, fine, get all the specialized stocks you want. If you are going to build one gun, get a general purpose stock. If you will need to pack or conceal the gun get a folding BC or some other folding stock.
4. If the gun is not to be used for long distance shooting at small targets, but for general plinking and pest control, get a small LIGHT red dot sight. The kids will love it, you will love it, and and any accuracy difference will be moot under the use specifications. If you want to hit small things out there at 100 yards or so, get a 3-9 with good glass. Weaver and some Simmons models come to mind. Get an A/O model.
5. For the kids Don't get a KID trigger. I have two. They are awesome. One is on my sil. gun, one is in the safe gathering dust because the other 7 10/22s I have work better for me with smooth, predictable, but heavier worked over factory triggers because I use them for plinking, action matches, and for loaners and for training people with a wide variety of gun familiarity and lack thereof. The KID is in a class of its own, but it is a small, specialized class IMHO.
6. What ever you do, enjoy the heck out of the gun. Oh, and build it, you will enjoy it more that way.