When it comes to modulating to different keys within the same song, there are ones that are more fluid than others. For example a modulation from the major to it's relative minor is a very fluid modulation whereas the modulation from C major to F# major is more abrupt. The further away the modulating key from the original in the circle, the more obvious and abrupt* the change will be. In most forms of classical music, there is a modulation as part of the form. For example the second movement is usually in a different key. You'll find that the composer would often follow the circle when choosing a key to modulate to. If they chose a key further away, if was often on purpose and for a good reason. So in effect, the simplest modulation would be from a major key to it's relative minor. One movement to the left or right would be the next easiest movement. The only exception to this would be the modulation to the major's parallel minor. For example going from C major to minor. Even though their key signatures are different, we hear this modulation so much, that it doesn't seem intrusive to us at all.
*Of course 'abrupt' here is subjective since accomplished composers can make the most unusual key changes seamless...or the most obvious change seem intrusive.This not only applies to composition but to improvisation as well. If' you're DJing you'll find the same thing happens when you mix songs. The further away the keys, the more obvious the transition will be.
The use of the circle is also a great vehicle for improvisation. Go through some theory books or jazz courses (the Aebersold series is a great example here), and you'll find that they'll often suggest going through the circle of fifths as an exercise. There are two reasons for this. First it's a great way to get the circle second nature in your mind. After going through the circle a million times with your scales, it becomes like your second name. Second, like mentioned above, a lot of modulations follow the circle so you're in effect practicing something that's going to come up in real world situations.
Try this: When practicing your scales, go through the circle. Play the C major scale up and down and then move to the right (or left) of the circle and keep going. You'll end up going through all keys in a very musical way. Try playing a pattern and doing the same thing. Next, try doing the same thing with chord progressions (e.g. a ii V I ). Since keys often modulate a fifth away, you're practicing things that will definitely come up. You'll find that the entire progression (a ii V I progression is all fifths) is just one movement of fifths after another.
We covered the circle and how it applies to chord progression before. Chords will often move in fifths. For example the famous 'Rhythm Changes' is just a chord progression going through the circle of fifths. Just like modulations, the movement of a fifth in chord progressions is a very pleasing sound to us. You'll find that the most well known ( and used) chord progressions (e.g. ii V I, IV I, V I, vi ii V I, etc.) are all just movements of fifths. At the same time, if you want to make things difficult, modulate across the circle. Start at C, then go to Gb, then to G, then to B, etc. If you look at notoriously difficult songs, (e.g. Coltrane's 'Giant Steps') you'll find that it follows these guidelines. Also, when improvising on changes in jazz or blues, it's common to add a V or ii V in the middle of the progression (again, just more fifths). These need to be on your fingertips.
Of course, all of this leads us to composition. Key modulation and effective chord movement and progressions are part of the craft. Knowing where to go, (or at least knowing where you want to go) is a huge part of effective writing. If you start off in the key of F and want to make a couple of modulations, what are your choices? Or, you're right in the middle of writing a beautiful melody and are trying to find a great way to harmonize that line, what are your chord choices? One of the things that you should be looking at in both examples is the circle of fifths. The 'smoothest' modulations are the ones that are closest to your home key on the circle. If you're right in the middle of writing a song and can't figure out the next chord, if it's not the root chord, try one a fifth away, you'll be surprised at how effective this is. This can also work in reverse. If you want to jump all over the place, if you want more angular changes or intrusive modulations, use the circle to pick the oddest modulation.
It's There, Use It
As you can see, there are tons of uses for the circle of fifths. It's best to have the circle on the edge of your fingertips. You'll be amazed how often you'll use it.