The parks will be super busy during your visit so I suggest you get there as early as possible. If you are staying on Disney property, find out ahead of time which day the Magic Kingdom opens early for you. (It used to be Mon, Thurs and Sat but please check for current info.)
Plan to GET to the park well before the scheduled opening. When it's crowded the gates may open sooner. Walk straight through Main Street, to the back of the Castle (I won't say through the Castle as it is sometimes closed off early in the morning.) Resist any temptation to take more than one picture of your family with the castle in the background right now
and don't look in any shops.
Behind the castle you will find Dumbo, Winnie the Pooh ride, Peter Pan and Snow White. I would do them in that order. The Carousel is there as well, if that interests your child. Dumbo is invariably the longest line and, as of July of this year, did not offer FastPass. Both Winnie the Pooh and Peter Pan offer FastPass so only get those if you want a second ride.
If Mickey's PhilharMagic interests you, head there next. It's a cute show but will take time and when you come out, ride waits will be longer.
Now, if the park is still somewhat tame, head to the Teacup ride. Usually not a long wait here in the mornings. From here, you have a decision to make. Do you want to head to Mickey's ToonTown? If so, check if your child is too small for the barnstormer (I think 3 is too small.) ToonTown may excite a 3 year old and may not.
Obviously, the child is too small for Space Mountain, Splash Mountain, Big Thunder Mountain, so my advice would be to head towards Frontierland. Only you know if your child will tolerate the Haunted Mansion well. It's dark and can be intense if you are 3, so I'd bypass that and go right to Aladdin's Magic Carpet ride (much like Dumbo but shorter lines). Then head to Pirates of the Caribbean. If you've done the Park early and in the order I've outlined, this will probably be your first place to encounter a real wait.
The one ride that your 3 year old will probably love and hasn't been covered is the Buzz Lightyear ride. It's near Space Mountain and does have FastPass so you may want to split, send you or your spouse with the 3 admission passes to get a Buzz FastPass while you head to another ride line.
The new Stitch's Great Escape ride just opened but I am not sure of the intensity so check before you travel to see if it's appropriate for a 3 year old. It is near Buzz Lightyear.
For meals, if your child likes the Winnie the Pooh characters, make breakfast (optimally, if not, get dinner) reservations at the Crystal Palace for the day you plan to do the Magic Kingdom. Best character meal in the Park (well, Cinderella's Castle is great but you are too late to reserve that.) I usually reserve for the latest breakfast (1030a?), that way you can knock out a bunch of rides before eating. If your child can do without Winnie the Pooh, pass on character dining while in the Magic Kingdom and concentrate on rides.
On days when you don't go to the Park, you can eat with the characters at 1900 Park Fare in the Grand Floridian. Make a reservation (we like dinner, as they have Prime Rib and a kids buffet). You can also get character dining at the Polynesian (Minnie's breakfast in O'Hana was nice) as well as fun breakfast dining (non-character) at Whispering Canyon in the Wilderness Lodge.
A pointer.....our kids always love Downtown Disney. There is no admission fee (though admittedly parking stinks). If you have a "down" day with no parks planned, head there early (10a)....it's empty. There is a great Legoland Store with tons of outdoor interactive areas and your child will be entertained for hours. There are also several large Disney souvenir shops to browse through, as well as a McDonalds, Ghirardelli and Rainforest Cafe amongst others, to catch a bite to eat.
As fasr as the Animal Kingdom, again, arrive early. Plan well and go straight to what you really want to see first (safari?). The lines at the Animal Kingdom get long fast so you'll want to work at a good pace. Your child might enjoy down time at the Boneyard play area. The AK is usually not an all day thing, so if you have the park hopper pass you can use it to spend the evening at another park, possibly watch the fireworks at Epcot or Fantasmic at MGM.
Have a great time....remember to get to the parks EARLY!