Why couldn't you slingshot around the sun and planets a few times to pick up speed before leaving the solar system? In the way the slingshot racer reached super high speed on his run to the gate in The Expanse?
The rocket doesn't know, but an outside observer will. If you're on the rocket, you experience a constant 1g force. The rocket doesn't "feel" that it's harder to accelerate, but an observer that remains at rest relative to the rocket's starting point will see it speed up less and less the faster it's going as it approaches the speed of light.
They continue to accelerate at the same rate for their frame of reference, but relative to earth or Andromeda their time moves slower and with their time moving slower their acceleration slows down to
It certainly going faster each time relative to the speed of light, which increases the mass. At first this is minuscule, but it starts to add up by the time you're going a decent fraction of c. It will asymptotically approach infinity, which is why nothing with mass can be accelerated to c. It also means that at some point, you won't have enough energy to continue accelerating.
