If a Jewish man and his Jewish girlfriend have sex regularly, what is the best way to atone for this? I know that it's kares, but is there any way to atone ?
When a person sins, he makes teshuva (repentance). Part of making teshuva is absolving not to perform the sin again.
If a person has no real intention of changing his actions, then his teshuva is pretty much worthless. As the Sages say:
With regard to one who says: I will sin and then I will repent, I will sin and I will repent, Heaven does not provide him the opportunity to repent, and he will remain a sinner all his days. (Mishnah, Yoma 85b)
God isn’t going to take you seriously if you ask him to forgive you and you have no real motivation to refrain in the future.
I have seen firsthand 2 cases of couples who lived together unmarried and became religious during that time. In both cases their rabbis had them move into separate apartments and only see each other in public for 3 months prior to getting married.
This is not a general rule, just the specific advice given in these specific cases, and is provided purely for informational purposes.