Take the free shuttle to the Happy Isles stop or just walk the extra half mile or so from Curry Village. From there you will follow the John Muir Trail (henceforth JMT) for 0.8 miles to the Vernal Fall footbridge and your first view of Vernal Fall. From here you leave the JMT and head up the Mist Trail proper, which follows closely along the Merced River (while the JMT takes a less steep path to the top of Nevada Fall). From the footbridge to the top of Vernal Fall is only about a half mile, but the climb is up steep granite steps. In the spring and early summer, it will be like hiking in a downpour due to the mist from which the trail gets its name.
The Vernal Fall lookout is about 1000 feet above your starting point and is usually crowded with hikers having lunch and relaxing. From there it's a short walk up to Emerald Pool which, depending on the season, may be a rushing torrent or a calm pleasant pool. Watch out here and keep tabs on your children. The rocks are quite polished and this has been the scene of many accidents. In the summer of 2005, a hiker walked out into the water a bit to fill his water bottle, slipped and got caught in the deceptively strong current, and was swept over the falls. Remember, you are only a couple of hundred feet from Vernal Fall and it is never safe to swim here.
From Emerald Pool, you cross a footbridge to get you over the Merced and then meander a short ways through the woods until you get a view of Nevada Fall. The trail stays a bit further away from the water and is not as steep as it is at Vernal Fall, but you get nice views nevertheless. At the top of the trail, there are composting toilets which you should definitely use in a high-traffic area like this. From here, take a right back toward the Merced and be sure to enjoy a moment at the Nevada Fall lookout. Because Nevada Fall is not truly vertical and the lookout sticks out a bit from the lip, this is probably the park's most beautiful view of a waterfall from the top. Not to be missed.
To round out your day, hike down the John Muir Trail which will provide you with outstanding views back at Nevada Fall on the upper part and a somewhat gentler, more knee-friendly descent.