For this answer I will use Ensnaring strike as the target example.
Is that strength save applied as soon as the target is hit, when it's the targets turn, or both?
Definitely the first one
The save is applied as soon as the target is hit. We can see from the description for Ensnaring strike (PHB 237, emphasis mine):
The next time you hit a creature with a weapon attack
before this spell ends, a writhing mass of thorny vines
appears at the point of impact, and the target must
succeed on a Strength saving throw or be restrained by
the magical vines until the spell ends.
So, on the same turn that the target is hit, they must make a strength save. If they succeed, then the spell has no effect at all.
If they fail, that brings us to the second part of your question:
If both, then does running the save on the targets turn cost its action since it has to make a skill check to get free?
As it says in the description for the spell (emphasis mine):
A creature restrained by the vines or one that can touch the creature
can use its action to make a Strength check against your
spell save DC. On a success, the target is freed.
So, if they fail the initial saving throw and are ensnared, someone (either the target or an ally in touch range) must use their action in order to free the target with a Strength check (not a Strength saving throw). Of course, they don't have to use their action breaking free if they don't want to.