You should visit a supermarket outside of the Disney area to get some essential supplies. This only applies if you're staying at Disney World resort hotel, because if you're staying at an outside hotel or Villa, you'll be doing this anyway.
There is a Super Target not too far from Animal Kingdom Lodge, which is to the west of Disney property. We visit here on the second day of every trip to get some supplies for our holiday duration. There are other places, like Walmart, but they're a bit further away.
Florida is hot, and it is very important that you stay hydrated. Although there are some places you get can free water in the parks, most of the time you'll have to cough up at least $3 per bottle of Dasani branded water. The cost of doing that sure adds up, especially if you're keeping a whole family hydrated. Even with just the two of us, we can see how much money we go spend just buying drinks at the parks. At Target, you can buy a pack of 24 bottles of water for around $7. Stick it in a backpack and put them in the fridge at your hotel.
If you've never seen the range of American snacks at an American supermarket before, you'll be in for a treat. The sizes of items are huge, and there is just so much. Disney only stock a few snacks at the hotel, and they're priced up, but at Target, many are really cheap and they have a massive selection. We usually get some for ourselves, as well as some for gifts...
The American candy is a great cheap way to give people gifts when you're back home. Target do stock quite a bit of cheaper Disney merchandise, too. We tend to get a couple t-shirts and mugs here.
If you've chosen to travel light and not pack many toiletries, you can go to Target to purchase some. They have a wide range of items and pretty much sell everything. They have some English known brands, as well as some funky ones.
If you've not rented a car, the only real way to get there is by a taxi. You can go to the front of your resort and they'll call one over to you. Depending on your resort location, it can cost anything up to about $20 each way to get to Target. To get back, you can ask the Target guest services desk to call you a taxi back.