Once you finish the main storyline, the endgame stuff opens up.
One endgame chain is the hardmode primals. You go talk to Minfilia at Waking Sands and get the quest for Ifrit HM. You have to go attune to a beastman aetheryte, then it is added to your DF. Ifrit HM can be done with AF gear, level 49 weapons, though better gear (wanderer's palace/amdapor keep drops, GC gear after you beat Amdapor Keep) is obviously beneficial.
Once you've cleared Ifrit, you get a quest to unlock Garuda HM, which is a similar process of tuning to an aetheryte. Garuda HM requires stronger gear, preferably Ifrit weapons and Darklight/AK/WP.
Once you have cleared Garuda, you can unlock Titan by returning to the Waking Sands once more. Titan requires even stronger gear than Garuda, and it is recommended you have majority iLvl 70 pieces (darklight, relic).
Wanderer's Palace can be unlocked by a female NPC behind the waking sands building. It drops the lowest item level gear of the 50 dungeons, which can be helpful replacing your AF to get ready for Amdapor Keep. It also drops the bloody materials for relic base weapons.
Amdapor Keep can be unlocked from Nedrick in Vesper Bay or through your Grand Company, and must be cleared for you to finish your GC promotions. It is a very difficult dungeon in AF gear with sub-50 accessories/armor/weapons, so you should spend some time in WP to gear up.
Many dungeons, I believe starting with Stone Vigil, drop Tomestones of Philosophy, which can be used to purchase Darklight gear in Revenant's Toll. There is no cap on how quickly you can accrue Philosophy stones, but there is a cap on how many you can hold. SV up, as well as all primal fights, reward you with a number of these stones.
Amdapor Keep and the Hard Mode Primals reward you with Tomestones of Mythology as well, which can be used to purchase AF+1 armor, which IIRC is top of the line. Myth stones are also used to upgrade a relic weapon.
Relic weapons are the reward of a quest chain, which can only be described as lengthy if you never played FFXI. You start it with Nedrick, then go to Revenant's Toll and speak to an NPC there, then to North Shroud to meet the blacksmith. The blacksmith asks you to find a treasure chest in a beastman stronghold, which will pop when you have the quest active, no levequest required. Once you've turned in that key item, he'll ask for a crafted level 50 weapon with 2 level 3 materia melded. The materials for this item can be obtained by high level gathering and from Wanderer's Palace. Once you've turned that in, you'll need to defeat Chimera, an instanced fight in Coerthas that cannot be Duty Finder'd. Once Chimera is defeated (and this is as far as I've gotten in the quest so I don't know EXACTLY offhand the order, apologies if I mess it up), you have to kill a large number of level 50 beastmen, then fight Hydra, an instanced fight in...Western Thanalan? The one with Camp Drybone and Halitali. After Hydra is defeated, you need to defeat all three Hardmode Primals. All these fights must be done on the job to which your relic belongs. Can't go on SMN if you're doing the SCH relic, for example, if you want relic credit. After you defeat the three primals, you need to purchase some items with Philosophy stones and your relic is complete!
The Binding Coil of Bahamut is a very high-difficulty dungeon that is intended to be challenged after Crystal Tower, a raid dungeon that does not currently exist in game. Because of this, successes in the Coil have been slim to none, although groups of very talented, dedicated players have beaten most of it at this point. Last I heard, no one had cleared turn 5 yet, though I don't keep track of the latest updates.