One of the most aggravating things to deal with when writing a song is coming up with proper transitions, usually between verses and choruses. However, transitions can also occur as key changes, progressions, and tempo changes. This is not an exhaustive tutorial on the subject matter but rather an overview of some of your options.
Regardless of what kind of change you are dealing with, you should always consider an intermediary change, or bridge. Yes, this adds yet another change and must now also be connected with, both, the preceding AND succeeding parts. However, the intermediary part is usually a lot less musically dense, so it should be easier to lead in and lead out of than simply conjoining two completely disparate parts. Sometimes this bridge is as simple as a stripped down drum break, or even easier, a 1 or 2 bar sweeping effect. That is usually the best option if the chorus (for example) contains elements familiar to the preceding verse. If your chorus sounds a lot like the verse, there is no sense in drawing-out the suspense and tricking the listeners ear into anticipating some sort of big event (which will likely be a letdown if it sounds much like what was preceding it).
In more extreme cases, like odd key, tempo or instrument changes you should consider somehow blending elements from the part coming up with the part that is just ending. In literature this technique is known as “foreshadowing,” which is to plant an idea in the observer’s head (perhaps subtly) in order to anticipate a future event. In music, this goes a long way towards getting the listener to accept what would otherwise be a sudden change. You could, for example, make a bridge that contains a simplified aspect of a new part upcoming in the chorus. If its a jarring guitar sound being introduced, for instance, then you can simply take the first guitar note played and lengthen (sustain) it, reverse it and stick that in the bridge.
As a general rule, the more extreme that the change is, the longer you should give the listener to anticipate and adapt to the change. Modern music-makers often want to shorten the “dead zones” in their tracks so they will often shy away from doing anything that they think is too boring to keep the listeners attention (no doubt a product of the the rapidly depleting attention span of modern ears). However, one forgets that the added time intervals we are talking about here are merely seconds long. Even today’s most jaded, multitasking, iPod skip-fiend is unlikely to switch off your track right between the buildup (verse) and payoff (chorus). And that’s probably the most important point: focus on the most attention-sustaining elements of your track and don’t dwell too much on the transition. Even if it bugs you to no end, it’s unlikely to kill the listenability of your track even if it’s not done well. When in doubt, keep it simple and keep it long (please, no “that’s what she said” jokes).
A good reference for this is the song “We Are Young” by the band Fun. The song features a very abrupt change (key, tempo, instrument, the whole deal) right from the first verse to the chorus, with no bridge at all. And yet that did not stop the song from reaching #1 on most modern music charts. Do you figure the band is still obsessing about that transition (or lack thereof)? Probably not, because they are too busy enjoying their newfound success regardless of that aspect of their song. Which goes to show that its still really the almighty chorus that counts.