Acceleration does not curve spacetime, only gravity does. The equivalence principle does not say that gravity and acceleration are the same, it says that if you choose a region of spacetime small enough that tidal effects can be ignored, then gravity and acceleration will behave the same way.

What happens if you keep accelerating in space?

If a space ship is using constant acceleration over interstellar distances, it will approach the speed of light for the middle part of its journey when viewed from the planetary frame of reference. This means that the effects of relativity will become important.

How does curved spacetime cause acceleration?

The curvature of spacetime (not just space) is responsible for gravity. Literally, near heavy objects, the “future direction” points slightly down. So anything that moves forward in time will find its trajectory pointing down slightly. This takes the form of downward acceleration.