Did you have the house inspected by a reputable home inspector before
purchasing it? I suspect not, because all this is something that could
have been easily uncovered. I'd get a home inspector pronto, to look
at not only this problem, but the whole house. It's virtually
impossible to diagnose something like this or give solutions without
seeing it.

I'd also take a good look at what's going on outside. Is the soil
graded away from the house? Are there gutters and are they working?
Is water from the leaders led away from the foundation as far as
practical? Get out there during a good rain too, to verify what's
happening.

If you're facing costly repairs on the basement issue, I'd contact a
lawyer, or consider small claims court, depending on the amount. You
may have a valid case against the seller, as it sounds like things were
done to hide problems that the seller knew about. I'd make sure to
take pics now of what happened, showing how it was concealed, etc,
before making repairs.