I'd recommend taking whichever you enjoy most to 50 first. And if you aren't really enjoying those two classes (Monk/DRG) enough that either stands out as 'wow I can't wait to keep going and hit 50 with this class and see how I can play with their full range of skills!' I'd suggest trying out additional ones to see if you enjoy them more. Right now if you're focusing on gearing your very first lvl 50 class in the highest level gear, the thought of purchasing entire sets of equivalent gear for other classes that can't reuse what you have already may seem daunting, but by the time you have them leveled you'll have more currency to spend.

-Poetics will eventually have no weekly cap, just like soldiery, if they stay with precedence with how Soldiery was capped at 450/week, and prior to that Myth tomes were.
-You can get all the WoD i120 equipment you want now, and can get i100 accessories via soldiery easily, and upgrade to i110 with ST drops, and unless you plan to start using your next class in Second/Final Coil right away that should be more than good enough I imagine. I100-i110 is fine for WoD, EX/HM Dungeons and such, especially if you play the class optimally. For Bard half my left side items are from WoD (and which is free and now unrestricted based on quantity per week) since I prefer their secondary stats so it's not like I had to spend all my Poetics on full left and right side, and same for my other level 50 classes.

My main point is I wouldn't let gear (in particular, being able to share accessories) be the main factor in deciding what to level to 50, as compared to which you enjoy more. It's probably the least important factor in the long run as earning tomes and getting gear drops gets easier after your first 50. Sure you could use your i120 or i130 STR accessories with DRG/Monk, but if you enjoy NIN/Bard more, you'll be able to afford Dex accessories by the time they hit 50 and you have between now and June to keep earning Poetics. But if it's more varietyyou want in playstyle, you could try casters out (if you haven't yet) and see how you like them. BLM and SMN can share a lot of gear with each other. As can WHM and SCH. For me, going from a Bard/Dragoon first, with all their Instant skills (with GCD 'recast' but no wait for initial use) to Summoner & Black Mage as casters were a bigger adjustment, but going between physical vs caster DPS classes gives me a much more different play experience than switching between one physical DPS class and another. If you don't like caster/healing classes, then no reason to play them, but I would recommend basing the decision on experimenting with the classes at low levels vs. expectations for sharing gear. I personally enjoyed leveling the Arcanist based classes, and getting access to both a DPS job (SMN) and healer job (SCH) at the same time while only having to level their shared class to 50 once (and leveling a DPS to 50 while getting Healer queues in DF is awesome).

A few people might say my opinions here are completely invalid because I haven't finished Coil yet (if so they're narrow minded)... that's not where my priority has been based on availability to commit to a static. The only exception I can think of to my advice to not worry about being able to share your highest lvl accessories right away, is if you're planning to get your next class ready for Second/Final Coil immediately after 50, in which case I'm guessing you'd need to be super highly geared off the bat to meet DPS checks and need to use those i120/i130 STR accessories... otherwise, for all other content you should be able to pick the class you enjoy best, and if stat isn't STR, buy Soldiery accessories to start with and upgrade over time and you'll be fine, that's what most of is with mult classes have had to do as we can't gear them all up at once either.