FFXIV Crafting-Combat Class Unlocks Useful to Gathering
It's largely a matter of personal preference. A lot of content unlocks at level 50 and again at 60, but it's mostly content that's only available to your level 50 or 60 job, so if you plan to then go back and level up others, you'd be taking a break from that content anyway. Some players like to get to the endgame content as soon as possible, so they focus on a single class/job first to get there quickly. Others like to spend more time on getting everything out of the earlier content, and level up several things in tandem. The game's really flexible about letting you level up as many as you want as far as you want. It's up to you when and to what extent you want to take advantage of that fact.
I'd recommend at least trying out all the classes at some point, but it's up to you when to do that and how far to take them. For whichever jobs you plan to continue into late-game (level 50 and up) content, though, you'll want to level their associated secodary classes far enough to pick up cross-class skills. For instance, once you get Pugilist to level 30 and Lancer to level 15, you can unlock Pugilist's job upgrade, Monk. As Monk, you can use cross-class skills from either Lancer or Marauder, so it's worth leveling those two (LNC to 34 and MRD to 26). If you want to advance other jobs as well, each will have its own set of secondary classes to level along with it.
Whether you decide to divide it between multiple classes or not, you should be sure to do the Main Scenario Quests (the ones with the flaming ring around the icon) as you get to them, as those unlock a lot of features. And every five levels, there's a class quest to take at your class's guild (for each class you're leveling).
Once you reach the level 15 Main Scenario Quest "The Ul'dahn Envoy" it will unlock the ability to take airships to the other two starting cities, so more classes will be available at that point. (Conjurer, Archer, Lancer, Botanist, Leatherworker, and Carpenter guilds are in Gridania. Arcanist, Marauder, Rogue, Fisher, Armorer, Blacksmith, and Culinarian guilds are in Limsa Lominsa.)
Either works. Gathering and crafting might get you more final fantasy xiv gil faster, but they add the extra effort of leveling the gathering/crafting classes. If you like the game's crafting system, that's the best way to make ffxiv gil, but if you don't, then you can make plenty from combat rewards and loot.
FATEs are good. There's also the hunting log, which gives each class its own set of mobs to kill for extra XP. Starting at level 15, you can unlock dungeons that can be run repeatedly for extra XP and loot, and those are probably the fastest way to level, especially with Roulette which puts you in a random dungeon or trial and once a day gives an extra XP bonus.
You get access to the entirety of the original areas at level 15, though it does take 'till 41 to unlock the Ixal quests. I can't think of anything the combat classes can provide at 50 that would help crafting/gathering.
Combat class unlocks useful to gathering/crafting:
Level 10: Ability to add additional classes (including the crafting and gathering ones).
Level 15: Airships, and with them access to all original (Realm Reborn) territory and classes.
Level 17: Retainers (These are NPC assistants who do various things for you, including storing extra items and selling items for you on the Market Boards. The first of those you'll definitely want for storing all the various materials needed for crafting, and the second lets you make ffxiv gil from your gathering and crafting.)
Level 20: Grand Company supply and provisioning missions.
Level 20: Chocobos (and the mount system in general). Not needed for crafting, but useful for reaching gathering locations quicker.
Level 41: Ixal beast tribe quests. (useful for leveling up crafting classes)
Level 55: Moogle beast tribe (like Ixal, they're crafting quests useful for leveling DoH)
I wouldn't suggest waiting to start crafting beyond maybe the level 17 unlock for retainers. Otherwise just keeping your crafting and battle classes relatively near each other in level allows you both to use crafting to make gear for your battle classes and use those battle classes to get crafting materials that drop from enemies and unlock leve quest locations.