3 Clarified that there is only one save and that it is mandatory. Highlighted difference between save and check
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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, optionally the second 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 canmust 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 checkStrength 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 checkStrength check (not a Strength saving throw). Of course, they don't have to use their action breaking free if they don't want to.

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, optionally the second 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 can 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.

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.

2 highlight the distinction between the saves and checks involved
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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, optionally the second 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 can 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.

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, optionally the second 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 can 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. Of course, they don't have to use their action breaking free if they don't want to.

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, optionally the second 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 can 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.

1
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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, optionally the second 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 can 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. Of course, they don't have to use their action breaking free if they don't want to.