The career plan that I've chosen doesn't start until next year.
You use the present perfect because the actual choosing of the plan isn't in focus: the fact that currently it has been chosen is the important point.
The grammar is correct now, but there is still a problem with the semantics. You've already made a plan, so it's wrong to say that the plan for your career doesn't start until next year. Perhaps:
The career path that I've chosen doesn't start until next year.
Student or Learner