When it comes to dungeons and tomes, it basically goes like this: run the high level dungeons you can, and obtain drops from them to gear up once you hit endgame. You'll want to hit areas where the gear is a definite upgrade, but not where you're sorely outclassed. So, starting out you'll likely want dungeons that drop level 60 gear. The libra eorzea, fansites like GamerEscape, and the play guide on the lodestone are handy ways to see what drops a dungeon has. Once you've got a smattering of upgrades, feel free to advance to the next tier.

Alternatively, you can hit the highest level dungeons you can, and skip some of the lower tiers. This CAN be faster, but people won't like to see you do these, and if other people had the same idea you'll likely end up just wiping repeatedly. Crystal Tower seems designed for this, with the first phase dropping ilvl 80 gear, the second phase ilvl 100 gear (and the items to upgrade soldiery gear), and having 24 people so your individual contribution is less.

Now, as you run dungeons (and certain other endgame events like beastman quests) you'll get tomestones. There are two, tomestones of soldiery give level 100 gear that can be upgraded to 110, and tomestones of poetics give.... I think it's level 120 gear that upgrades to 135. They're also used to obtain weapons of their respective tiers, but need an extra item to do so. The item for soldiery drops in Syrcus Tower and the second coil, while the item for poetics drops in final coil and presumably will also drop in the final phase of CT. Anyway, use tomestone gear to upgrade slots you haven't really gotten drops for, and you can eventually use them to fully gear up in almost the best gear.

Now another thing to note is your relic weapon. You can start a quest for this once you beat the main scenario, and continually upgrade it throughout the game. Reaching level 90 is easy, and should be a great weapon to start with. Past that, it gets rather grindy and it's up to you whether to keep going, or get weapons from primals, tomestones, or final coil.

Primals drop various non-armor items. The hard mode of the first 3 and good king moggle mog drop weapons. The extreme modes of those and ultima hard mode drop accessories. Leviathan, ramuh, and shiva all drop weapons from their extreme mode versions, and the latter two drop rings as well. At the moment, only ramuh and shiva don't drop things you can find better of elsewhere for easier, but the others can be useful for stepping stones.

Finally, binding coil is 8-man dungeon content with the hardest fights, and drop armor and accessories from each turn of a given coil, and weapons from the final turn of each coil. At the moment, only the second and final coils are relevant, but you WILL want to be well geared first, at least full level 90 for second coil, and full level 110 for final coil. Still, this is currently how you will get the best gear.