Do you have some experience in tanking or healing in FF14 yet? If not - I suggest that you play your Monk first and enjoy the new content with this class.
I am not fan of this "this new class looks cool so I want to try it out"-philosophy if you have no experience so far in this certain role.
Of course, leveling all classes is something you can do, but as for now (doing the new 3.0-story and reaching lvl 60) do it with your main, then.
The new jobs look quite complex and they will only frustrate you - and that's not worth it, imo, when new lvl 50+ content is waiting for you. After reaching lvl 60 you're, of course, free to level up any other class.

I will level DRK first - but I am part of a serious progress static and I'll level with 3 other players of it. All with roles they actually going to play once Alexander is open (I'm their OT so I need war/drk anyway till then). So if you don't have this kind of ambition and just want to have some fun start with lvl 50 right away.