Get on the Osceola Parkway from the airport. Heading towards Disney World, you'll come to International Gateway. You can't miss it - the huge Gaylord Palms will be on the left on the other side of International Drive. Make a left onto International Drive. You'll be heading towards Highway 192. Before you get there (not that far down), there will be a Publix on the left that should have everything you need. There's also a Walgreen's on the right if you need a drug store.
After you are done shopping, return to Osceola Parkway and continue straight. You'll basically run into Animal Kingdom Lodge at the end.