What does a Terminator do after its mission is accomplished?

A Terminator is essentially given a mission or an objective ("Go back in time and kill Sarah Connor" or something of the like).

What does a Terminator do once its mission is accomplished?

We see in Terminator 2: Judgment Day (the live-action movie) that the T-800 (played by Arnold Terminator Schwarzenegger) completed its mission of protecting the Connors, and then sort-of just melted itself into nothingness (presumably to protect the world from discovering Terminators again).

But what would the T-1000 or the T-X (in Terminator 3) have done?

Are they given multiple directives ("Go kill Sarah Connor, then go kill Robert Baratheon, and then have a nice hot bath; you've earned it")?

This question is related, but not a dupe: What were the Terminator's instructions following termination of Sarah Connor?

Answer

Terminator

The original script treatment makes it clear that the Terminator did have a way of telling whether it had got the right Sarah Connor. It was checking for a distinctive leg injury she'd suffered prior to Judgment day. The irony is, of course that she suffered that injury while fighting the Terminator:

Vukovick stops the report. Did he hear correctly? Two homicides in one day with the same name?

"That's not all that's the same," Buckman says, lifting one of the girl's pant-legs which has been slit up past the knee. Also slit, from ankle to knee, is the skin and muscle of her calf, peeled back like a hotdog bun to expose the shin-bone.

Vukovick scowls. The same mutilation as the Encino housewife, left leg only. Too fucking weird. The news guys'll have a field day with this... the first one-day pattern killer.

Since we know that the Terminator has a "win condition" and cannot self-terminate, this gives us two possible options for what it would do after; proceed to any secondary targets or simply hibernate.

T2: Judgment Day

In the novelisation for Terminator 2, we're explicitly told that the termination of Sarah Connor is a secondary target, even after the death of John Connor.

At the Voight house, in John’s bedroom, the T-1000 was reading the last of the letters from Sarah. It scanned the return address on the envelope, PNT-82, ISOLATION WARD, PESCADERO STATE HOSPITAL, and the date, (only two weeks ago), and quickly concluded that the primary target may go there. In a matter of moments it was moving down the street, away from the city, and toward its secondary target.

Terminator 3: Rise of the Machines

In Terminator 3, we see that the T-X has a list of "secondary targets"; known accomplices of John Connor who would become his key lieutenants in the fight against Skynet. In the absence of a primary target, she begins killing the secondary targets. It's reasonable to assume that the Terminator would also have some tertiary targets, perhaps known rebel strongholds or senior commanders that it would attempt to terminate.

Terminator: The Sarah Connor Chronicles

In the episodes "Heavy Metal" and "Self Made Man" we see a Terminator place itself into a low power hibernation-mode in order to avoid polluting the timeline. Given the evident restriction on 'self-termination', it's possible that the Terminator would just hide itself until it could make itself available to Skynet.