The first step will be to build up your Allagan Tomestones of Mythology. You can purchase ilvl 90 equipment with that and while there is other equipment that beats it, it's a good first step. The most consistent way to do this is by running Duty Roulettes and receiving the once a day bonus of tomestones. At this point, you should have the Main Story and Low Level categories unlocked. You should also concentrate on unlocking and completing other post-Main Storyline dungeons to unlock the other Roulettes.
It's probably best to do those dungeons with FC mates or friends for the first time. My suggested order for doing the dungeons that will unlock Duty Roulette: High Level:
The Wanderer's Palace
Amdapor Keep
Haukke Manor (Hard)
Pharos Sirius
Halatali (Hard)
Brayflox's Longstop (Hard)
Copperbell Mines (Hard)
The Lost City of Amdapor (Hard)
I would suggest being upfront with groups about being new to the dungeons if you end up using Duty Finder to enter them, rather than going with a group of FC members or friends. The reason being is that a large portion of the playerbase likes to go through them quickly, and that would be very difficult if your equipment is of a lower ilvl.
Doing the following dungeons should unlock Duty Roulette: Expert, which gives you a boost of Allagan Tomes of Soldiery (the newer tomestone that can be used to purchase ilvl 100 gear). You probably shouldn't do these until you have finished your Relic Weapon quest.
Hullbreaker Isle
The Stone Vigil (Hard)
The Tam-Tara Deepcroft (Hard)
In between working on unlocking Duty Roulette: High Level, you should be picking away at your Relic Weapon quest. You will need to do some of the easier dungeons in the High Level Roulette group, the Hard Mode Primal fights, the Chimera fight and the Hydra fight. They should be relatively easy to get through since the addition of the Echo buff and Duty Roulette: Trials bringing people in. Depending on how much gil you have from earlier, you may need to save a bit to pick up the base weapon used in the Relic Weapon quest; I have been pretty lazy with my relics and bought them off the market board with the appropriate materia already melded in.
An alternative route that can net you tomestones and allied notes is by participating in Hunts. You should unlock the ability to participate in Hunts with your Grand Company. If you join a Hunt group, you can get tomestones quicker than dungeons, but it may be less consistent than dungeons. Your daily hunts will provide gil and allied notes, but at a slower rate than if you join a Hunt group; daily hunts don't provide tomestones. I wouldn't recommend spending allied notes on gear you can buy with mythology, but it is an option if you really want to get that gear quickly.
Depending on what your other job levels are, you might also want to look at leveling Lancer/Dragoon. Invigorate and Blood for Blood are very useful for Monks (and other non-magical dps).
Watching some videos before doing some of the dungeons and fights can be helpful. I personally like MTQcapture on youtube, but I know some people swear by MrHappy.
I hope this helps!
EDIT: "City" not "Cit".
Edited, Sep 25th 2014 9:49am by Illsaide