Bring your toddler, preschooler, or little one to Yosemite! There's plenty for your family to explore.  A list of things to do follows. 

But first, a couple of things to keep in mind for your child's safety:

1. Wildlife is wild and unpredictable.  Do not try to feed any wildlife; this includes squirrels, chipmunks, deer, and birds.  Also do not try to attract squirrels for pictures. They do bite, and can carry diseases.  Deer may appear tranquil, and you will see people getting within arms reach of them. This is illegal (yes a park ranger can give you a ticket) and extremely dangerous.  

 2. Keep a sharp eye on your child around water in the park, particularly in April/May/June but year round as well. The water is extremely cold, and may look calm from the top, but probably has very strong currents that are beneath the surface. Also, the rock around rivers becomes extremely slippery even when a little wet.  Be careful when rock hopping near streams/rivers/waterfalls.  Stay on the trail. Definitely do not let anyone wade or swim at the top of Vernal Fall, no matter what you see people doing. Again, this is illegal and extremely dangerous.

THINGS TO DO

Stroller friendly options in Yosemite Valley:  Mirror Lake, Lower Yosemite Fall , Bridalveil Fall, Cook's Meadow.  Check out this link to find trail descriptions, and locations.  Though the first two trails will be dry from late July through the fall, they are still lovely walks even without water. They provide nice views, and good family time.  Happy Isles is not listed on the link below, but it is a very short stroll near the river beginning from shuttle stop #16.  Follow the signs for Happy Isles.  In the winter, be aware that all of these trails could have patches (sometimes large patches) of ice.  Use caution and keep your eyes out when on the trails.
http://www.nps.gov/yose/planyourvisit...

Glacier Point is a drive from Yosemite Valley (approx 1 hour).  It is a paved viewing area that lets you look down on Yosemite Valley and out to some higher points.  Great option when the road is open (usually late spring through late fall.)  Hold on to your child's hand or strap them into the stroller.

Other meandering strolls:

  • Wander behind the Ahwahnee (shuttle stop #3) out to the river.
  • Walk from Yosemite Lodge (shuttle stop #8) on the bicycle trail to Swinging Bridge (it doesn't swing) for a nice view of meadows, the river, and in the spring/early summer Yosemite Falls. Though this area is a popular swimming area, just don't let anyone jump from the bridge...it is illegal and dangerous. 
  • From Degnan's Deli (shuttle stop #4) turn left on the bicycle trail, and walk 20 minutes on the flat trail to the Ahwahnee Hotel. Nice views of Half Dome and Glacier Point.  At the Ahwahnee Hotel (shuttle stop #3), hop the shuttle back to whever you want to go.

For the more adventurous: Some younger children have lots of energy and don't mind hiking.  Four year olds go up to the footbridge of Vernal Fall. Know that it is 1.5 miles roundtrip, and it is uphill all the way to the bridge. That section of the trail is paved.  Do take plenty of water and snacks with you. Some young children do go further up this trail and continue up the steep granite, slippery stairs to the top of Vernal Fall. Just keep a sharp eye on children, as this trail can be extremely slippery, and the stairs are big, even for adults.  See the valley hikes link in the stroller friendly options section.

Indoor options: The visitor center at shuttle stop #5 and #9 has exhibits that can be touched and listened to. Behind the visitor center you'll find a self guided walk though an Indian village (stroller friendly), with different displays, a roundhouse, and other traditional structures.  Next door to the visitor center you'll find an Indian Cultural Museum with a touch section for kids, as well as a demonstrator working on different traditional projects. At shuttle stop #16, you'll find a nature center usually open from late spring through early fall.  

Guided Programs: Children of all ages are welcome on virtually all programs in the park. However Wee Wild Ones is the best suited for the 6 and under crowd.  Topics rotate. It's a highly interactive and fun program.  One hour Junior Ranger programs generally run late spring through early fall.  The programs handle a wide age range.  Four, five, and six year olds tend to enjoy the program the most.  This age group may also enjoy Ranger Ned's Big Adventure, a live theater program for kids.  Adults must remain with their children at all of these programs.  

Activities: from late spring through early fall Yosemite Lodge and Curry Village rent bicycles and usually rent bicycle trailers.  Virtually all of the bicycle trails are flat.  Avoid riding on the road, though it is techinically legal. Drivers in Yosemite tend to be very distracted.  For your safety, it's best to stay on bike trails only. 
http://yosemitepark.com/Activities_Bi...

In the summer, there are stage coach rides in the Wawona area at the Pioneer History Center. Check the Yosemite Guide newspaper, in the Wawona section, for more information.
http://www.nps.gov/yose/planyourvisit...

In the fall, there's a Halloween program at the Happy Isles Nature Center (typically the Saturday before October 31st). The Nature Center is open in the evening and rangers tell stories and organize night walks. The best part: Glowing carved pumpkins lighted with candles along the path to the Happy Isles Nature Center.  For more information, check the Yosemite Guide.
http://www.yosemitepark.com/Activitie... 

In the winter time, Badger Pass is a great area for young children to learn to ski.  Badger Pass has a history of developig very specific programs for very young children.  The ski area is also very small, making it very family friendly.  Tubing is also available.

http://www.yosemitepark.com/Activitie...

Little Cub Program: The equivalent of the Junior Ranger program geared for younger children. Buy the book, complete 5 activities (some coloring, picture drawing, etc.) and kids receive a Jr. Ranger badge as well as a Little Cub button.

National Park options for kids

EATING

In the Valley, Yosemite Lodge has a food court that can pretty much satisfy anyone's picky eating habits. They now have child size hamburgers (plain burgers, condiments on side, fries/drink) as well as grilled cheese sandwiches.  Cheese pizzas are available.  Soup, chili, spaghetti, and other pasta are kid friendly options.  The Valley has two pizza areas.  From late spring to fall the Loft above Degnans Deli near the visitor center serves pizza.  Curry Village has a pizza deck that is open year round.  There are many other options, but these seem to be particularly family friendly.

During the high summer months Glacier Point sells hot dogs, chips, sandwiches. During a similar time frame Tuolumne Meadows has a grill that is open until 5 pm.  

CAR SICKNESS

Yes, the roads into (and inside) Yosemite are all windying.  If someone has a really big problem with car sickness, Hwy 140 is the lowest elevation road, and it is still winding but the curves are much more gentle than the ones on Hwy 120 or Hwy 41.  

NOT RECOMMENDED

In the spring through fall, an open air tram does a tour of the Valley (tickets required).  Some young children to enjoy the ride, but keep in mind it is 2 hours long with one bathroom break (outhouse).  A park ranger is talking virtually the entire time.  This tends to stretch the attention span of most toddlers and preschoolers.  If you visit in the summer and decide this is an activity for your family, try to take the 10 am or the twilight tram (5 or 6pm-ish) to avoid the super hot temperatures.  

FINAL ADVICE

Take a read over the Yosemite Guide newspaper before you visit.  It breaks down the park into manageable chunks.   In the guided programs section, any programs printed in colored font (as opposed to black) are geared specifically to kids.

Carry plenty of water (not just juice) and snacks with you for the trails.   Sometimes and impromptu picnic under a tree is the best activity of the day.