How far could you go in space in one direction in a rocket if you had an eternity of time

Would you ever hit a boundary that would be the end of space?

If yes, what would be on the other side of the boundary?

As far as science knows you would travel infinitly. We do not know if you would come to the end of the Universe and then enter a dark nothingness or if the Universe is infinite at which you'd continue on seeing stars, galaxies, and all there is in space. Or, what if, like planets, you found yourself, after millions and millions of years, back at planet Earth proving that the Universe is round but we are stuck on the inside.

The rocket would keep moving (Newtonian physics works quite well in these types of situations) until the craft encounters something that gets in its way or gets snagged by a gravity well.

No, as the void between galaxies is expanding at a faster speed than you could feasibly travel at.