First, see if you can try to match the original hardwood. A flooring specialist should be able to help with that. I'd carry that onto the stairs, hallway, and master bedroom. For kids rooms, go with a durable, neutral carpet. It will save you some money plus it's softer for them to play/fall on.

If you CANT match the hardwood or can't afford to add more hardwood, I would just put newer neutral carpet upstairs. Faux wood never comes off looking good against real wood and while most people hate carpet in living spaces, they're perfectly fine with it in bedrooms/upstairs areas.

That hardwood looks to be red oak with either oil based finish that has oranged (ambered) over time or a light stain. The SPECIES of the hardwood is pretty easy to match. If you are going to add more hardwood, then the SPECIES match is the most important thing. You can always work with a different COLOUR on the upper floors. That is completely acceptable.

And as Sina points out, you can always do the hallways/master bedroom in the hardwood and then run carpet (durable as you can find) in the other areas. Again this is perfectly acceptable. Many, many people prefer carpet in bedrooms and offices.

Because the hardwood looks to be red oak in a 2" - 3" wide plank, this should be super easy and (relatively) cost effective to find/install. You can save money by adding carpets to the other areas.

Red Oak can then be added to other rooms at a later date if you feel you want a house that is wall to wall hardwood. It is one of the easier species to find/install/match. They key to this is the species match.

It seems like you are trying to go for a more clean modern look as you mentioned a clean grey color. I would go with laminate through the entire house and change up the color, going away from the original flooring. Maybe something a little darker or even ashy. I have two young girls myself.. and I honestly believe carpet is a bigger pain to clean. Crayons and spills can be mopped but carpet stains easily and shows it’s wear much more than laminate or traditional hard wood. Good luck!

Thanks for the advice. I think we'll get a flooring person out to
evaluate and try to carry through the red oak and see what happens from
there.

IMO since you already have a lot of wood I would continue that upstairs and as mentioned a good floor pro will be able to match pretty darn close , add a runner to the stairs to break up the 2 floors. I just noticed those bookshelves in that one space those should go bye bye IMO unless this is a must for book storage. Before going too much further i would look into adding lighting .I like solid floors for kids with area rugs for some softness and make sure they can be easily cleaned.