Answer: When modulating to a new key there are tried-and-true techniques, like "common-chord" modulation, in which you move to a chord that is known to both keys and its function just changes as the new key takes over.

F and Bb would be particularly good candidates for modulation, being a fifth apart. F is of course the tonic chord in F major, but it's also the dominant chord in the key of Bb, so it is common to both keys. A particularly strong change would be to get to the tonic at the end of the F major part - that will happen anyway - and add a seventh to it, producing F7. F7 won't seem like the end any more (no one ends a piece on a dominant seventh chord), so the listener will now expect something else to happen. And what will happen is Bb, the resolution of the F7 chord. Modulation from any key to the one a fifth lower (or a fourth higher) almost gives the impression that the entire previous music was just preparation for the new key. You want smooth; it will be as smooth as a gravy sandwich.

Here's an example: