What was Big Willy Shakes going for?
Adios. Sayonara. Adieu. Ciao. Auf Wiedersehen.
Biondello is simply saying goodbye. This one is as straightforward as it gets. All the guy means is that he wished Bianca goodbye indefinitely. He's going forever, and then some, so a really long time. Just like he says: there's nothing more to say after that.
Shakespeare had more to say though. He used "forever and a day" again when Rosalind and Orlando talk in As You Like It.