How can I transfer items between my characters? Will this work if they are in different leagues?
Two characters in one league
Your stash is shared between your characters in one league. You cannot transfer quest items in this fashion, all other items are fair game.
From hardcore to standard
Create a character in the source hardcore league, place an item in the inventory, die. The character and its inventory (not the stash) gets transferred to Standard (not to league softcore).
Transferring items between other leagues is not possible within the game.
From challenge league to standard
Wait until the current challenge league ends (usually every few months). Then the contents of all your stash tabs in the challenge league are transferred to remove-only copies of those tabs in standard.