Hi Erin,
My recommendation for a moderate hotel for a family with young children is
Port Orleans Riverside, both for the convenience of travel and the amenities. It was the resort where we stayed on our first Walt Disney World vacation with children many years ago, and it remains a favorite.
The resort is centrally located; the bus ride shouldn't be more than 15 minutes to any park. While the resort does have more than one stop, I find the transportation there less time-consuming than at the Caribbean Beach or Coronado Springs resorts.
The adjacent Port Orleans French Quarter has a single bus stop, but often shares a route with Port Orleans Riverside. While I love French Quarter and often choose it for adults-only weekends, I think Riverside has more to offer children. As a guest at Riverside, you still can use the pool at French Quarter with its sea serpent slide and grab hot beignets from food court.
Port Orleans Riverside has six pools, including the main pool on Ol' Man Island, where you also will find a fishing hole and a playground. You'll also be able to participate in a number of family activities, including bicycle rentals, poolside movies and campfires. One of the bonuses of the Port Orleans resorts is the water taxi service directly to Downtown Disney, too.
In terms of accommodations, lots of the rooms at this resort come with a bunk-sized pull down bed that is perfect for a child. (If yours are like mine, they don't like to share a bed). Or you can stay in a Royal Room here, outfitted by Princess Tiana to host visiting princes and princesses.
I hope that helps. Please let me know if I can answer any other questions for you as you plan your visit. Enjoy!
Wilma