My family went to the Y several times growing up and my siblings and I all worked there during summers in college.
The Y offers day camp for kids if you want to drop them off and let them spend some time making new friends and have some time away from the kids. On property they also have trail rides, mini golf, indoor pool, craft shop and a lot of other activities. The property has a lot of space to get out and "explore" if you aren't wanting to venture into RMNP everyday. Lots of big boulders for the kids to climb on.
The park will require reservations if you want to drive in, and parking can be a hassle if you aren't there early. But you can walk into the park and get on one of the hiking shuttles from the Y grounds.
If you want to hike bring water/snacks and try to make it fun for the kids. If you make it fun they won't realize it's a hike and that they walked for 3 miles. Count trees, jump from rock to rock, red light green light, have them try and find the "perfect pine cone". If they're bored they'll complain.
Bear Lake is one of the main parking lots and trail heads. It is less then 100 yards to the parking lot and from there you can go on to some other lake hikes or Alberta Falls. For an all down hill hike for the kids you can go from Bear Lake to Bierstadt Lake and use the shuttles to get back to the car or park at Bierstadt, shuttle to Bear Lake and then hike back to the car.
Sprague Lake also has a parking lot and boardwalk around it. Moose can be there pretty regularly in the mornings or evenings.
Trail Ridge road is fun to drive and get up above tree line, lots of scenic overlooks and a gift shop/visitor center at the top.
Moraine Park has a small museum/discovery center that is fun to explore with kids.