Where to start in End-Game content? (Paladin)

[Kyana]

Should I start farming Castrum for Phils and get full Darklight?
Should I start working on getting my Relics?
What set of accesories should I start with? Darklights or dungeon drops?
Allagan or iLv90 sets? What should I aim for?

You can start your relic quest and do the first steps but it would be wise to upgrade a bit before starting with Garuda and Titan.

What I would do:

Run dungeons for tomestones AND gear upgrades.
Run duty roulette for more tomestones if you want. You can limit the selected dungeons by not unlocking them.
Once you have gathered more party experience, watch a video or read a guide about the hard mode primals. It helps if you roughly know how the fight works when you attempt it for the first time. The rest is practice.
Since you are tank, your damage is not THAT important, but it would be good if you at least had Ifrit or Garuda weapon for the fight against Titan.

You can also buy or craft ilvl70 gear and meld materia in it if you want. (HQ and with materia it's better than DL)
And you should run Crystal Tower for more tomestones and ilvl80 gear, which is quite good btw.

For Coil, a good start is having relic and ~ilvl75.
By the time you can start doing Coil you will probably have some tomestones of mythology which you can spend on a few pieces of ilvl90 gear.

Allagan or myth gear doesn't make a big difference.
Don't wait for Allagan drops if you have tomestones of mythology to spend.
Get the mythology items for the slots which drop in Coil T4-5 first, because it will be some time till you get these drops.