Most songs stick to one key, but I've noticed that switching keys is sometimes barely noticeable (to listeners) while playing guitar.
In which circumstance would changing your key mid-song sound alright?
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There are various techniques in theory for when key changes sound good (look at circle of fifths for example) but in reality you can make almost any key change sound good in context.
The simple examples are those you get in pop songs - where a final chorus or verse shifts up a step or two (the so called Truck Driver Gear Shift) but you do see some fine examples where it works really well (there are various collections of these - have a listen.)
It all depends what you are trying to do at the time - so my advice to you is to just experiment. Sometimes it will work, and sometimes it won't, but you'll learn a lot about context along the way.
It's really just like anything else in music. Some things "sound right" and others don't. Furthermore, things that "sound right" to some people sound very wrong to others. So, what's important is that they sound right to you. If some people like what you're doing and others hate it, it's probably pretty good. (If everyone hates it, either you're a genius who's years ahead of his time or you have work to do.)
Modulation between closely related keys (keys a fifth apart, relative major and minor, for example) are pretty common, because they are pretty easy to do. You just play a chord common to both keys and move on as if you were in the new key. Try playing these chords: Am, E7, Am, Dm, F, G7, C. That moves very smoothly from Am to C, Am's relative major.
This Elton John piece has a pretty striking modulation, from Eb minor to G, which is a pretty distant key (six flats to one sharp). But it works, to my mind.
Finally, here is a nice introduction to various kinds of modulations, with some examples.
There's generally not time within a pop/rock song to do much with structural key contrasts. It's fairly common to jump to a remote chord to begin the 'middle 8', but it's often only to give a platform from which to immediately work your way back home, maybe through a 'cycle of 5ths' progression.
Note that my answer is full of hedge words - 'generally', 'fairly', 'often'. There's the 'three chord trick'. There are mini-symphonies. And there's everything in-between.
When a change is 'barely noticable', we could question whether there's really a modulation at all. But we could argue all day over what's a modulation, what's a brief excursion, what's just a chromatic chord or two. Recognise what's happening, don't worry too much about categorising it!
I think a key change can often work well if the lyric has a change in energy or a strong change in the point of view. The key change should align with the lyric change, though. If you want to amp up energy, you will probably want to either go upwards in pitch, or cycle upwards on the circle of fifths. Relaxation in energy would be the opposite direction, usually. Large changes in key signature (many accidentals changing) are more likely to give a sense that you've gone somewhere pretty far from where you were, and small changes seem nearer by. These are just some general guidelines. Have to use your ear and feel the implication of the sounds, if that makes any sense.