There are many different ways to modulate from one key to another, and without hearing your specific group I have no way of knowing what they are doing. This is why I recommend you listen to what your group is doing, and try to learn to recognize the chords they are playing (presumably there is someone leading the group -- possibly just one of the other instrumentalists who everyone tacitly agrees to follow). Worst case, I'd recommend asking someone ahead of time what they're doing.
In my experience, most actual modulations occur rather quickly (within a few bars in Classical music, and often immediately in pop music, with no intermediate chords), so the "quite a few bars" that you are describing seems unusual. Until you get to atonalism (and you aren't) there's not really such a thing as "in between keys" (at least not for more than a chord or two) -- you're pretty much in either one or the other (or a different one).
What might be happening is that most of this section is either in the original key, with a modulation just before reaching the new key, or the modulation comes early on, and there's just some progressions in the new key. Another possibility is that they make multiple modulations to one or more intermediate keys in between, before arriving at the final key. If you want a longer chord progression, when in doubt, the circle of fifths always provides options for smooth chord progressions, by always going to the chord whose root is a fourth higher (or fifth lower) than your current one. Picking chords a third lower than the current chord also tends to give pleasing results. If possible, you may want to focus on using chords that exist in both keys (this isn't always possible, depending on what chords are in each key). Of course, you could always do weird stuff with augmented sixths, or diminished sevenths, but given the nature of the music, I very much doubt this is the case.
At some point, by the end of your chord progression -- and this is the most important part -- you're going to want to establish the new key by using a dominant/tonic progression (a cadence) in the new key. This means playing the V (or V7) chord followed by the I chord in the new key. This is possibly what you are noticing when suddenly realize that you are in the new key.
As you can see, there are really far too many options for me to provide you with a very useful answer. But since you ask for an example, here's an extremely simple way to transition between two chord progressions in different keys, using only a single new chord. I'm going to slightly "rotate" your first progression to something I think sounds a bit more natural, and then move into the second, modulating from C to G by using the dominant 7th of G, which is D7.
|: F - C - Am - G :| F - C - Am - D7 |: G - C - D - Em :|
Part of the logic here is that the original phrase ends on the dominant of the original key (G) so we just modify it to end on the dominant of the new key. Another helpful fact is that the preceding chord (Am) leads into the D chord really well, because D is a fourth above A (look at the circle of fifths: A -> D -> G is just moving around it) and because Am is a chord that exists in both C major and G major.