In Diablo 3, I buy stuff in the auction house but I don't know where to find it
You have to send items purchased from the auction house to your stash explicitly. This extra step is so you can purchase things you don't actually have space in your stash for, it lets you queue up to 50 things in the auction house "outside" of the game.
You do so from the left side of the Completed tab.
Select the item purchased, then hit "Send To Stash".
Pictured here is sending gold from the sale of an item, but the same system applies to buying items.
It doesn't really make sense that you have to send gold to the stash IMO, but whatever.
Go to the 'Completed' tab in the Auction House and check the menu on the left-hand side. You can send items to stash from there (and gold from completed auctions, too!)