It's good to keep in mind that it takes a lot of practice to learn an instrument. You have to dedicate a lot of time and it will take years to become a good piano player (same for all the instruments). So, if you want to learn how to play multiple instruments, you will have to practice all of them. If you don't have a lot of time, it will be hard to practice them at the same period of time.
This doesn't mean you cannot. A lot of people practice their skills on multiple instruments.
Would becoming a good piano player in any way hold me back from playing another instrument professionally?
Of course not. Quite the opposite, in fact! I started with double bass for a few years and then I started playing the piano and the fact that I already knew rhythm, melodies, harmony, theory etc, made it easier to learn the piano. If you already know an instrument, it's going to be easier to learn another one. To oversimplify it, you just have to learn the technique in the new instrument.
So my advice would be to start with the one you want to start with (I guess piano in your case) and then if you have the time, take up another one. But, I would also suggest that you take up the second instrument after you have learnt quite decent piano, because otherwise, it'd be quite difficult to learn from scratch two totally new instruments when you don't know anything about music.