How do I use the shared stash to transfer items between my characters?

The stash box is located in the northwest corner of Sanctuary, just past Zed's clinic. To get access to the stash, talk to Claptrap (who is in the same location) and he will give you an absurdly "impossible" mission. Follow the mission to its conclusion, which will point out the position of the stash. You can deposit up to 4 items in the shared stash, which you can remove with any other characters you create.

See also this walkthrough video on IGN.

Aside from the stash box mentioned in the current accepted answer, in True / Ultimate Vault Hunter Mode, you can also access Claptrap's Secret Stash near the starting location of the game.

It is found at Claptrap's Place, Windshear Waste (in a closet where several broken claptraps are stored and the first "Cult of the Vault" symbol can be found).