What's the goal of each Adam and Claudia?
It seems to me that Adam would like to end time which sounds like he would like to break the cycle while Claudia would like to keep it?
Adam wants to destroy the knot, wiping out both Adam's and Eva's worlds. He plans to do this by killing the pregnant Martha from Eva's world and her unborn child (The Origin / The Unknown) by using the energy of the apocalypses of both worlds on her.
Adam [talking to Eva's world Martha in S03E06]:
What's growing inside you, the origin, must die.
But it can't be killed by normal means.
It's born of both worlds, and the energy of both worlds is necessary to destroy it.
The apocalypse in my world, and the apocalypse in your world. This here is the end.
A machine that not only crosses time, but also space.
It focuses the energy of both apocalypses onto this one point.
Your son only exists because the matter exists.
And now he will die through the matter, and all of us with him.
None of this... will exist anymore.
Both worlds will extinguish one another. Absolute annihilation.
Adam's plan, however, doesn't work.
Claudia's goal, on the other hand, is to save her daughter, Regina. She learned about the original world and that it is the only world where Regina doesn't get cancer.
Claudia [talking to Eva's world Claudia in S03E07]:
I still remember exactly what she said.
"If everything goes right... Regina will live."
I've thought about that all these years.
I just can't believe that by that she meant her suffering was to repeat over and over forever.
There must be a way to untie the knot without destroying all life in it.
A way for Regina to live.
I think neither Eva nor Adam know that path.
But I'll find it.
In my world or yours.
Claudia [talking to Adam in S03E08]:
I've spent 33 years looking for answers in your world and in hers.
I've tried to put together the pieces of the puzzle.
To understand how everything can be reborn from the same family tree over and over again.
Until I realized that we're not all part of the knot.
Both worlds are a cancer that must have grown from something else.
If you remove it, you destroy all that was born of it, but you keep everything alive that already existed in the origin world.
Claudia [talking to Tronte about Regina in S03E08]:
She isn't part of the knot.
She will live.
Claudia [talking to Adam in S03E08]:
Everything I did, I did for Regina.
All these years, I've been searching in both worlds for a way for her to live.
A way to break the chain of cause and effect.
Both in your world and in Eva's world, I've tried to understand how everything is connected.
Until I finally understood that our two worlds were borne of another.
That Regina will always die in both worlds.
That she can only live in the world from which everything was created; the origin world.
Claudia [talking to her younger self in S03E08]:
If everything goes right... then Regina will live.
Progress with the time travel portal is slowly moving forward. The setbacks start to take their toll physically and Jonas is forced to accept one after another. He is no longer worried about preventing the apocalypse - he wants to destroy the knot and wipe out the world. Jonas has become Adam.
Adam holds Martha from Eva's world captive. He knows she is pregnant with the origin of the time traveler family tree. He ties Martha up in the time travel portal in the former nuclear power plant and directs the forces of both apocalypses on her, killing her and her unborn child. Adam believes this will destroy the knot and end his own existence, but his plan doesn't work. He's still alive.
Old Claudia arrives and explains to Adam that his attempt has failed due to quantum entanglement. Eva found a way to change the course of events, meaning there is a second pregnant Martha out there that will give birth and preserve the knot. She explains that there is another way to break the cycle: by changing things in a third world, the original world where everything began.
In the summer of 1987, Claudia receives a visit from an older woman in her office The woman claims to be Gretchen's owner, which is true seeing as the woman is indeed an older version of Claudia. She was the one who brought Gretchen from 1953 to 1986. The woman shows Claudia the time machine and explains that she must stop a man called Adam if she wants to save her daughter Regina.
Claudia eventually realizes that neither Adam nor Eva have plans to save Regina so she takes matters into her own hands. When she meets Claudia from Eva's world, she shoots her and takes her golden time travel sphere. From then on, she pretends to be Claudia from Eva's world. Eva gives her the blueprints for the time machine that the clockmaker will later build.
Claudia tells Adam about the existence of a third world - the original world - which has been broken into two parts due to an unfortunate event. She tells him that it's possible to undo the event, ensuring Adam and Eva's worlds would never exist. Claudia is sharing this information in her own interest, because the original world is the only one in which Regina doesn't get cancer.
Adam wants to destroy the knot. He figures that the only way to do it is by killing the pregnant Martha using the energies of both worlds. However, for this, Adam needs everything to proceed in an untouched manner so that he can kill Martha at the right time, once she has conceived. But when he does kill Martha, nothing happens to the knot because an alternate pregnant Martha is alive and well thanks to Quantum Entanglement.
Claudia wants to end the knot too. But given the cancerous bootstrapped nature of both worlds, she figures there must be a third normal world which gave birth to the two knotted worlds. She knows she cannot convince Adam because he believes his plan is gold. So she waits for him to fail. Once he does, and is now prepared to listen, Claudia tells him that he needs to send Jonas to get Martha and go to the Origin World to destroy it. Adam agrees.
Adam's World actually has two Claudias and that is discussed here - How did Claudia come back from the dead?
The question is currently unanswerable as the show has been sufficiently vague about what's going on, with additional hints of ulterior motives and things not seeming what they are; enough so that you simply cannot tell right now.
Adam has been talkative but we cannot know if he's been truthful. He has repeatedly stated that he wants to break the time loop, and that he believes the "next" loop (i.e. the one after the events we witness) will be the last one. He's been mostly painted as the villain - but there have been subtle suggestions that Claudia's position is not as do-gooder as the show would have us believe - Adam mentioned this to young Jonas in 1921.
Claudia seems to be driven almost solely by wanting to counter Adam's plans. That doesn't prove she's the good guy in this conflict though. She's adamant and not much is known about her plans. Whatever Adam's true intentions are, it seems reasonable that Claudia is aware of them and wants the opposite. But even that isn't fully clear - maybe Claudia doesn't know Adam's true intentions and they may want the same thing but not realize it (I don't think so, personally, but it's not been disproven yet by the show).