While it is certainly possible to main all three of those, I wouldn't really call it effective. You can generally swap out any one DPS for another without affecting much, so having multiple different mains of the same role doesn't increase your versatility by squat. It just means you have a little more choice in how to play. Ideally, having a DPS, a tank, and a healer as mains is nice, as you can join any party (Minus very particular ones that want exacting setups)

As far as leveling strategy goes, I feel you should start off with ONE main job. Healers and tanks get better queue times for the duty finder, but you can certainly do this with a DPS as well. When you hit level 30, level up whatever sub class you need to 15 in order to unlock your job quest, and you may wish to do a little research into what cross-class abilities are considered vital for your job and level up other classes to get those as well. Other than that, press onward to 60 with your chosen job. The reason for this is to unlock more things to do. Glamours, events, travel options, gilmaking opportunities, having a level 60 class opens up a lot of doors for you that will make leveling subsequent things easier.

Now, that said, once you decide to level up other things, it generally doesn't matter if you level them up to 60 one by one, or level them up at the same time, with a few minor exceptions. The first is if you level two classes that are in the same category: Fending (Gladiator, Marauder, Dark Knight), Healing (Conjurer, Scholar, Astrologian) Casting (Thaumaturge, Arcanist), or Aiming (Archer, Machinist). These classes share almost all their gear, minus their weapon and certain job-specific armor sets. Leveling these up at the same time means you can share gear between them, potentially lessening the cost to level. The other reason to level things up in tandem is if you're also leveling gathering and crafting classes: Outside of a few hiccups early on, generally the system is designed such that a level 20 gatherer and level 20 combat class can get materials that a level 20 crafter can use to make level 20 gear. Be warned though, making all your own stuff can be time consuming!