what to do after landing in Amsterdam at 8:30am

If you can't check-in early...see if you can leave your bags at the hotel (or lockers at train station)...then go to breakfast. Then go for a walk about the town. Amsterdam is easy walking or biking because it is very flat. Depending on the day of the week...you may find outdoor markets or street entertainment. Or take a canal boat ride. Then check-in and everybody take a nap. Then get up again...go see some sites...have some light dinner and try to go to bed Amsterdam time.

"not sure if we will have had much sleep on the 10 hour flight or not. We are traveling with two young kids."
The probable answer to your first question is: no, you will not have had much sleep.
So I would plan accordingly.
Jet lag, with 2 young kids, is probably going to be a challenge.
First priority will be trying to find some place to get rid of your baggage, hopefully the hotel. That would be a biggie.
Then find out when the earliest you can get into your room is.
Walk around, keep 'em moving, until you can get into the room.
In the room, you can clean up.
But try not to let the children (or yourself) go to sleep until evening, otherwise they will be up in at 2am and so will you.

anne,

depending on where you will be staying, i would drop off your luggage and walk around town. you can check out the various sights, museums, canals and shops.

chances are you won't get into town until 1000 or so.

There should be museums and shops open by the time you drop off your luggage. If you care to, check out the NEMO. Its like a giant ships bow and green near the main train station. Its to the right and its a big science center. The Maritime museums is to the right of the NEMO, if any of you are interested in such things.

happy trails.

You and your kids will be tired and cranky after a long flight and little (or no) sleep. Go immediately to your hotel and drop off your luggage. Then I suggest you take a one-hour canal cruise (many companies have boats outside the Amsterdam Centraal train station). That way, you get to see the city and not exhaust yourselves more by walking around too much. If you still need to kill time after the boat ride, the kid-centric NEMO science museum isn't too far from the station and should make them happy. Or, if you hotel is near it, head to Vondelpark and relax. There is a children's tree house nearly hidden in the woods, about midway through the park..

With young kids head to the Zoo. The Nemo science museum is nice but IMO jet lag and science exhibits aren't a good combination:)
Another possibility is the Keukenhof Gardens. It opens at 8am, there are direct buses from the airport and free lockers are available. Like the Zoo you will be out in the fresh air and there are lots of activities for children, like the petting zoo, arts and craft, birds of prey shows etc.

Wow some great ideas here! The boat ride sounds leisurely- any recommendations on which one? The lockers at the gardens may be great too if we were to go there. And I love the tip on the hidden treehouse in the park! Thank you!

The problem with a leisurely boat ride, IMHO, is when you're jetlagged and not active you get tired. I'd recommend dropping off your luggage and going for a long walk. Amsterdam has canals and great architecture. Go to a museum or maybe the Anne Frank house if it's on your itinerary. Try and stay awake until after dinner to acclimate better and maybe take melatonin when you're ready for bed.

I would suggest checking with your hotel. You may be able to check in early, either for free or for a small fee. They may not be able to provide an answer until shortly before your travel day, but it's worth asking. With two tired and possibly cranky young children, a nap earlier rather than later is usually a good idea. You could then go out after they've had a short nap and do some of the activities that of been recommended.

In my experience outdoor activities are better on an arrival day than indoor activities.