If you choose both modes, they happen in the listed order and nothing happens in between them. Notably, if the second mode targets an Aura currently enchanting the target of the first mode, that Aura will be returned to its owner's hand. (It wouldn't be put into a graveyard as a state-based action until after Run Ashore is done resolving.)

For the first mode, the owner of the card chooses whether it goes on the top or bottom of that library.