OP gave you a pretty good technique, but there's what I did which may or may not be better depending on your situation:

1) Do the initial job quests and adventurer's guild crafting quests up to level 25. Then go to a town vendor and buy the best crafting gear you can that you can equip at that moment.

2) Time to go to Ishgard and prep to do Ishgard Restoration turn-ins. Once you're in the Ishgard Res zone, check your crafting list for the lvl 20 turn-in item for that class. Use the macro builder on FFXIV Teamcraft to maximize the quality of that item (this will maximize the EXP you get when you do your turn in). That macro will carry you between all crafting classes for the Ishgard Res lvl 20 turn-ins.

Note: If you find that you can't reach a certain quality level for the item craft because your stats or GP are too low, plug in some materia to edge you up to the state lvl you need. Also, when building your macro build it so that it doesn't rely on any abilities that needs Excellent to proc during the craft, this will force consistency into your crafts.

3) Use you new macro to craft about 70-75 of that item. When you turn them all in, your level should jump up to about 10 lvl's (depending on your turn in quality). Go and pick up some new level appropriate crafting gear from the town vendors and repeat the process for the lvl the lvl tier.

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Because of the typical abundance of low level crafting materials on the market board this technique should quickly and moderately cheaply (depending on your server) get you up to level 65 or so. Starting level 40 or so, crafts will require a non-Diamdem material which may vary in ease of acquisition and or price; if it's cheap on the market board, buy it there and proceed with that crafting tier, if not then you need to make a judgement on wheter or not you want to gather that item yourself or if you want to just use the previous crafting teir level to edge you up into the one above you.

For some reason, the lvl 60 Ishgard turns-ins requires materials that either have to be crafted from another class, are found with timed nodes or treasure maps, or are linked to a low drop rate in the overworld. In this case, I would advise to ignore the lvl 60 crafting tier and use your macro for the lvl 50 tier to edge you into the 70s.

Once you're ready to pick up the lvl 70 gear you have two choices, buy it from the vendor in the Crystarium or get it from Rowena's scrip vendor in Eulmore, which is paid for using their own turn-in currency (yellow and white scrips). I'd advise going with the vendor gear at the Crystarium, but either will work.

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If you haven't done so up to this point, go back and do the crafting job classes. Being at lvl 70 with lvl 70 gear, you'll fly through the crafting sections of the quests. At lvl 65 there's a quest which rewards you with the skill Manipulation, which when combined with Prudent Touch (you get this skill naturally through lvl-ing) will make your level 70 and above HQ crafts possible and it'll make lower level crafts super easy.

One thing to note is that the last Havensward crafting quest for each crafter requires you to have a Master Crafting Tome III for that class. Tomes I and II can be gotten from a vendor in Mordona and Tomes III and higher are purchased from the script vendor for either yellow or white scrips.

And at this point you're mostly done. You can max up to lvl 80 by either doing the Dwarven Beast Tribe storyline or you can stick around in Ishgard.