Firstly, don't conflate or confuse between scales and keys. The two are connected, but generally we write in a key, and use the scale notes basically from that key diatonically - but often other notes tend to creep in too, so it's not made from a scale any more.
Several ways work and are used regularly. Moving to the key either side of the existing one in the circle of fifths is subtle. Say we're in key C, moving to key F or key G involves changing only one note diatonically, so the other six still get main use.
Using the parallel key works well, because there's still the root note, or home. So, C major into C minor - or vice versa.
On the same tack, going modal, using the same root has been known to do the job. So, C major into C Mixolydian, etc.
As above, using the relative key is good, as most original notes are retained. So, C major into A minor.
Introducing diminished harmonies (which are usually not 'in scale'!!) is also a time-honoured device. Because diminished chords are not major nor minor, they have that certain uncertainty about where they might lead - a perfect opportunity to escape from the existing key.
There's also the 'truck driver's gear change - simply put, suddenly (or subtly with the dominant chord for the new key) changing up a semitone or even a tone. So, C major into C♯ major, or C major into D major.
There are several more ideas - why not study some works, and even pop songs, to find out other ideas?