So much of what you ask can be boiled down to either personal preference or $ available, which makes it almost impossible to give an "answer". I can relate some of our experiences and preferences, which might be applicable since we went when our twins were 2 1/2 and again this year at 4 yrs old, but realize that you mileage might vary.
First, I have never really found a substitute for staying on-property. For us, it is all part of the Disney "experience", as I am sure they mean it to be. Our kids really enjoyed Pop Century on our most recent stay, with its larger-than-life theme. Remember that with young ones you are likely to want to go back to the room for a bit in the afternoon (hey, even us older ones were glad for that afternoon break). You'll also want to bring or rent a stroller. We had rebozo slings when ours were 2 1/2, and made out quite fine, but rented a stroller this last time. At 4 they were just too heavy to carry around.
The food courts at the on-site properties all open well before the parks, and there is quite a selection. You can eat it there or take it back to your room, though it can be a hike. Not much breakfast-wise in the park last I looked, but later on there is all sorts of stuff available from vendors and counter service places... no restaurant required. For the most part, when they were younger, our kids ate off our plate or shared a meal. The main meal expense for them was their milk.
2 days at Magic Kingdom is good for young ones. It allows a slower pace that keeps everyone happy. Not sure about 1 day at Animal Kingdom though (despite my reference above to slower paces). We tend to spend a half day there (meaning 1-2pm or so... then an hour break or so at the room) and then go to Disney Studios the other half, even now that they are older. There just isn't (or wasn't) all that much at Animal Kingdom for 3 yr olds. Disney Studios has a few things that our daughters really enjoyed, among the Voyage of the Little Mermaid, PlayHouse Disney, and Beauty and the Beast on Stage. There's also Fantasmic in the evening: though it was a little intense in spots for little Tori, she ended up cheering for Mickey at the end! You'll need to check on times for all that if you decide to go that route, especially in the off season as some may not be operating then.
Can't really say much more about it, except that our girls have really enjoyed it the two times they have gone. It was really fun this last time, simply because they remembered so much from the first trip. Don't let anyone tell you that 3yr olds don't remember, 'cause they most certainly do. And those memories are what it is all about, since they get to keep theirs longer than I get to keep mine!