If you could get a car just for the day, I'd highly, highly recommend Wallace Falls with the kids. The best part of this group of falls is that you can go as far as you want and will still see some good stuff, so if the kids get tired you can tap out and still have a nice day. There are officially the Lower Falls, Middle Falls, and Upper Falls, but there are other falls along the way as well - IIRC seven in total if you were to go to the top. Best bet with the kids would be to stop at Middle Falls, which gives the best view anyway and avoids the rooty, more difficult (though not that bad) part at the top. Would be about an hour drive from the heart of Seattle.
Wallace Falls On the way back, the Sultan Bakery is a ridiculously amazing place for lunch, breakfast, dinner, whatever. You would definitely need a car, though.
Also needing a car, and not a hike at all, Snoqualmie Falls are probably the most spectacular falls that are nearby - 35-40 minutes away but you'll still need a car. Not a "hike" as you can basically walk up to them, but seemingly beautiful (I have not gone there).
Another easy hike, close to Seattle (40 minutes), is
Twin Falls. If you're getting a car and going to hike, though, I think Wallace is a better choice. Just throwing this out there as another close-by option.
In other words, to see any cool falls, then yes, no car is a deal breaker. With a car there are several reasonable options.