My wife Karen and I took our three children on family backpack trips starting at age 6 months (we carried them) all the way through high school, at which point they decided it was more fun to go with their friends instead of the old folks. By age 3 they were hiking by themselves carrying nothing except maybe a favorite small stuffed animal toy in a tiny backpack so they could imitate mom and dad (who was laboring under a very heavy pack :-). They liked hiking into a lake no more than about 3 miles in where they could play (with supervision!) in the shallow sandy or muddy areas and we could take some short day hikes to other locations. Coming from the Bay Area, we had a different selection of close areas than you do in Carson City, NV, but there is some overlap.
A favorite area of ours with little kids was the Grouse Lakes roadless area in the Tahoe National Forest just off I-80. I would think that would be no more than 2 or 2.5 hours from Carson City. This is a highly glaciated granitic area with low relief and lots of small lakes, at a lower altitude (7000 feet). It gives the feel of the high country, but with much easier access. I like to start at the Carr Lake trailhead off the Bowman Rd on the west side. Take the Hwy 20 exit off I-80 towards Nevada City. About 5 miles down the hill, just before the road levels out in a big meadow, Bowman Rd takes off to the right (north) and is signed for Fuller Lake. You go beyond Fuller Lake on paved road about 20 minutes to a signed gravel side road to Carr Lake. From this trailhead, you hike a short distance
on a closed road to Feely Lake and then on trail past Feely Lake over a low rise to Island Lake - total about 2 miles. Here's a link that shows the current trails:
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Island Lake has many good campsites scattered around it and shallow areas to play in. Because of the lower elevation, this lake gets tolerably warm. You can take side hikes to other lakes - I recommend going north to the small Crooked Lakes, which have even better swimming. This is not a designated Wilderness, so there are no camping or fire restrictions (other than possessing a free California campfire permit - pick one up at any Forest Service office in CA, such as the one at Big Bend off I-80 about 5 miles east of the Hwy 20 exit). The trails here are open to mountain bikes, but I have never seen one at Island Lake - they prefer the more rugged route from Grouse Ridge east to Glacier Lake. Also, this area can get crowded with family backpackers or just folks from Sacramento who want an easy trip. Mid-week would be better than the weekend.
Another possibility off I-80 which we have done many times recently on day hikes would be Salmon Lake south of Big Bend also in Tahoe National Forest. This lake is set on a granite bench overlooking the North Fork American River canyon area, with the interesting Loch Levin Lakes only about a mile away to the east. But this is only good for kids if you come in from the western trailhead south of Huysink Lake. That way is only a couple of miles and a few hundred feet elevation gain. The main trailhead at Big Bend that goes first to Loch Leven Lakes has over 1000 feet elevation gain. To get to the Huysink Lake trailhead from the east, continue west on I-80 past Big Bend, past Hwy 20, to the Yuba Gap exit. Go over the freeway to the south and in a couple hundred yards, where the road splits, take the right fork towards Lake Valley Reservoir. This road is paved. Just before the reservoir, which has a campground, boat launch, etc., a high standard gravel road takes off to the left. It goes around the north side of the reservoir and then climbs up the slope beyond, briefly crosses the ridge back to the Yuba River watershed, then loops around to Huysink Lake. Ignore all the minor side roads along the way. Continue driving about 1/2 mile past Huysink Lake to a obvious trailhead in a flatter area. The trail is in good condition and easily followed. It's main destination is Loch Leven Lakes, so after a couple miles you have to watch for the side trail south to Salmon Lake, which may not be signed. It's on a sparsely vegetated granite ridge. Be sure to take your topo map. Here's a link that shows the trail:
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Again, this is a roadless area like Grouse Lakes, not Wilderness, with no specific camping or fire restrictions. Loch Leven Lakes can get extremely crowded on weekends with campers and day hikers, but this western trail is practically deserted and there are fewer campers at Salmon Lake.
I also second the suggestion to backpack to Winnemucca Lake south of Carson Pass. This is a short trail with a moderate amount of uphill. As a bonus, it has amazing wildflower blooms. As I recall, there is no camping right at the lake, but we found a good camp on the benches to the west. Continue on the trail just past the lake and then start looking up on the benches to the north a hundred yards or so off the trail. You can also make a nice short side trip to Round Lake with great cliffs on one side. These lakes are rather cold and deep for kids to play in, however.
When my kids were a little older (5), they also made it from Wrights Lake on the west side of Desolation Wilderness about four miles up to Twin Lakes, which have shallow areas to play in and dramatic alpine scenery for the adults.
-Phil