I would still do Metro because you'll spend a lot more time trying to find a parking space and getting your kids in and out of it than you will getting your kids on and off Metro. Plus, they think trains are cool. If it's a small stroller, it can fold up and go up the escalators. If not, most of the Metros have a working elevator. I raised twins here and we Metro'd everywhere with a big double stroller until they outgrew it. Even with a broken elevator, it was a lot easier to fold it up and haul it up an escalator than dealing with parking.
As far as museums, Natural History is great for kids. The Building museum near Gallery Place was awesome when mine were young but I honestly don't know if that's still the case. Things may have changed when they started charging for admission.
As far as restaurants, I'll let someone else answer that one but Embassy Suites is walkable to a lot.