Congrats!! I don't think there's a specific "cost"; as APowers noted, they can be as cheap or as expensive as you want. So this is an excellent time to think about what you want for your kids, and then do the research to figure out how much those things will cost.
The big one, obviously, is college. I assume you have thought about what you want to offer, if anything, because that seems to be the first thing parents think of. ;-) So other things:
- School: school supplies are not negligible, and the kids tend to destroy backpacks/string bags (not surprised, they have so much stuff in them there's a ton of wear and tear) -- most we have gotten is two years out of a backpack. But there are also a lot of fees and extras I didn't consider -- PTA fundraisers (yours and your friends'/neighbors'), gym/band uniforms (and "spirit wear"), field trip fees, hot dogs at football games, club/activity fees, plus associated costs from those activities. For ex., my DD did the speech and debate team, which involved 2-3 weekend tournaments + nationals. Note that all of this is optional; the question is really what you want your kid to be able to do. Also, by the time they hit HS, they can often get part-time jobs to cover part of the costs.
- Sports: our rec leagues run about $100/season, plus whatever gear the sports require (basketball = nothing; baseball/softball = cleats every year or two, a new glove/bat/helmet every 2-3 years, because the stupid buggers insist on growing!) -- but there are often places that sell used equipment from everyone else's growing kids, too. YMCA swim classes are about $60/6 weeks, plus @$75 monthly family membership. Karate was around $130/mo. We limited kids to one activity at a time (plus swimming until they passed our safety standards), more for our scheduling sanity than for cost, but I also know families who filled their afternoons and weekends with practices and lessons and such. Travel teams are both more time and a lot more money, including frequent out-of-town weekend tournaments; our kids aren't that talented, and we weren't interested in the lifestyle, so that wasn't even something we considered. So this can be as minimal or epensive as you want.
- Music: our kids start band/orchestra in 5th grade, so consider instrument rental costs (we got lucky: our schools are so desperate for low brass that they give kids the use of an instrument free!). We now pay $35/week for lessons, as DS has surprised us by actually being pretty good (but we waited until 8th grade to start, because I'm not going to pay for lessons if I need to nag him to practice). There are also periodic costs to audition for things like all-county band.
- Technology: our kids' school now loans all of the kids laptops, but before that, we got DD a cheap (@$400) laptop as a birthday present one year. We gave both kids phones right around MS -- for our convenience more than theirs, because we both work in offices, and I wanted them to be able to get in touch if they had a problem home alone after school. And then of course there are the gaming systems . . . . DS is now totally besotted, and even though we make him pay for his own stuff through his allowance or extra chores, we still need to come up with that cash.
- Daycare: these costs continue until MS or so! It does get cheaper, of course, but as late as 5th grade, I was still paying @$500/mo for before- and after-school care. And I also paid a teenager to babysit DD in 6th grade until we ran into a problem and I realized DD could manage herself.
- Driving: the other big one. Expect about a 50% insurance hike for a teen driver. And of course the driving lessons cost a few hundred bucks, unless you can get it through school. Plus whatever you want to cover for costs of car, gas, etc. (for ex., I agreed to pay half of the gas, since DD spent noticeable time running errands for me).
- Clothes: we probably spend $600-800/yr at Kohl's/Target, depending on how fast the kids have grown. I totally splurge on tennis shoes (not included in that total) because of some genetic foot issues. You can obviously do much better than this if you want.
YMMV. But that's the point: you're going to have different kids than I do, and you're going to want different things for them and have different resources and priorities. I'm pretty spendy on mine, because we are already FI and have free cash and I can afford to be (plus I grew up poor, so I enjoy offering my kids opportunities I didn't have, like sports and instruments and such). You're doing the right thing to plan for these costs, but first you need to figure out how much of the above you want to do, and at what cost/involvement level. Good luck!