We went in early May. It was a great time to go, not yet unbearably hot and not terribly crowded. We went with our three boys, 6, 3 and 19 months at the time. We stayed at the Disney Old Key West resort and really liked it. It is one of their older properties so the rooms were still designed on the large side. Grandma joined us, so we rented a 2 bedroom unit and had around 1200 square feet for the six of us, with a fully equipped kitchen allowing for meals in, and a washer/dryer, so I didn't have to pack as much. This particular property gives you a lot of space, a nice pool, (not as fancy as the newer resorts, but totally fine for kids in this age group) and since it's a bit older it's less expensive. From our experience, it is very well maintained and we'd stay again. If you stay there, get a unit on the first or second floor unless you don't mind stairs, since there are no elevators-and in one of the buildings closest to the main pool, and you'll be set!
It is more costly to stay on property, but we wanted to do this for our first trip for the whole "Disney Experience". We are glad we did. There are added perks to staying on property as you are connected to all parks via the bus system, which is really great. If you go during a less busy time the buses are not very crowded and really make getting to and from the parks/Downtown Disney much easier. If you drive, staying on property allows you free parking at all parks as well. There are also several parks to which you can travel by water taxi or monorail, which is also included. Another thing we liked about staying at a resort vs. private home rental (which we did look in to and it is much less expensive) is the swimming pool. Most private homes have one, but the resort pools just have that extra "wow" factor that the kids love, and believe it or not, even in Disney World, our kids did just want to spend time hanging out in the pool! One way to save money if you want to stay on property is to consider renting your unit directly from a Disney DVC (timeshare) member. We did this, and the cost was 1/2 of what we would have paid if we booked directly through Disney. There is a book out there, which I think is actually an online book that you pay for and download/print, called The Ultimate Disney World Vacation Guide, by Beth Haworth. Lots of money saving ideas and links to great coupons. This is how I learned about renting from DVC members, and exactly how to do it. Worth every penny, just for that.
I agree that with kids 6 and 3, the Magic Kingdom is the park to concentrate on. Great attractions, good layout, lots of characters. If they are fans of Winnie the Pooh, book the character breakfast at the Crystal Palace-awesome! Do book a character meal someplace, for some really great one-on-one character interaction at your table. The kids loved it! I was excited about Animal Kingdom, but was disappointed. The layout makes navigating difficult, especially if crowded. Our kids became bored rather quickly. The dinosaur area of that park was fun for them, though. MGM was boring for our kids, as would be most of Epcot-except there are a few attractions at Epcot they enjoyed (Nemo theme). We did park passes that allowed "park hopping", which adds a bit to your price, but is a great option with kids this age. If you want to spend just a few hours at one park, but then the rest of your day at another, park hopper allows for this. Buying passes for multiple days does actually cut down on your cost, so that is the way to go.
In hindsight, with kids our ages, we'd have spent more time at Magic Kingdom, skipped MGM entirely, except for the laser show at night (park hopper allows you to just go to a park to catch a show) and would have allowed for more relax time for the kids at the pool, which they loved-and needed, to decompress from all the excitement of the parks. Downtown Disney is also a fun place to spend some time shopping, eating and relaxing. Do your souvenir shopping here-all the same stuff you'll find at the parks (maybe more) and you don't have to carry it all around with you all day!
Have fun making your plans, and good luck. Whatever you do I'm sure you and your family will love your trip!