Our family has been doing a similar trip each summer for 7 years (girl 11, boy 8).
I agree with everything said about renting the big RV (sleeps 8 if you are friends) and heading to Cooper Landing. Camp at the Russian River CG or Quartz Creek. If RRCG you will want to advance book, I believe it is 6 months on Recreation.gov. Late July is the 2nd run of reds (AK fish and game has historical run charts/daily counts on their website) We take the kids up the Russian and avoid the lower river - 15 min of walking and it is nice fishing for kids. Great fun to go below the falls, ferry the kids across on your back and fish the opposite bank with no one else. There should be some good trout fishing starting on Quartz Creek and the Russian. Too early for the crazy trout fishing on the Kenai, but some good red fishing if you can get away from the crowds. Guided, Paul at Alaska Angling Addiction or Dusty at Trout fitters are my favorites, but everyone will have their own.
With a troop of kids and people who haven't been to Alaska before I highly recommend the trek to Homer and a full or 1/2 day halibut trip. Homer is stunningly beautiful and the kids have a great time catching halibut. We do the 1/2 day with Rainbow Tours every year, which is about half wildlife sightseeing (sea otters and whales generally) and half halibut fishing. Fish are not big but kids catching 10-15lb halibut one after another is great fun. Seldovia is a great day trip on the ferry if it is sunny. If you decide you want to see bears, Homer is a good departure point. It is expensive ($600+PP). They generally go across the Inlet to the coastal side of Katmai. You could split the group if some people really, really wanted bears and were willing to spend the dollars.
If you wanted to do Katmai/Brooks Camp that generally means a flight to King Salmon ($350), $200 float plane to Brooks, and camping (if you booked a site) The lodge is supposed to be $$$. It is really incredible but I get my fill of bears in about 1/2 day. At that point unless you have a guide you are limited to fishing the Brooks river. Great place to fish - completely amazing really - but a short river full of bears and almost as many park rangers.
Seward is a good day trip from Cooper Landing. Well worth it. If it is first timers, I'd highly recommend the glacier tour boat. I don't think the rest of the world has anything quite like it.
For us 10 days with Denali looks something like this:
1. Arrive ANC, get RV, Get Groceries, Drive 5 hrs to Denali
2. Take Bus through Denali, leave as early as you can manage...
3. Get up late at Denali and drive 5 hrs back to Anchorage (we have friends, so "camp" at their house, catch up, let the kids get reacquainted and play, and arrange for cataraft we share to be delivered to Kenai later in the week)
4) Drive to Homer (5+ hrs). We leave late... Could fish along the way and still get to Homer by 10PM. Camp on Spit at Municipal CG.
5) Homer - 1/2 day Halibut & Pratt Museum
6) Homer - Seldovia trip (could be bear trip)
7) Drive up to Kenai - maybe combat fish in or around Soldotna... Stay night at Quartz Creek.
9) Seward (could split group, some do more fishing, some do Seward Tour and Sealife Center) 60ish minutes from Cooper Landing to Seward.
10) Back to ANC, Drop RV, Go to Airport
Obviously you can compress several parts of this plan and make trade-offs. Our primary "drop" is Denali. Picks up 3 days on the Kenai/Russian. We've done it, rode the bus, saw the mountain, bears, moose, etc... If people haven't they will want to. It costs days of fishing.
If you can go back country, off the road system, absolutely do so. Nothing more fun for experienced outdoors people than a float trip on a river full of salmon, but it means committing to one type of "experience." We generally have to budget $2Kpp for gear, food and airplanes to pull off a good 7 day float (have done it for 800pp, but that took a lot of insider effort and fuel was cheaper then).
We may see you there. Haven't decided whether we will go first or 2nd run this year.