I'm new to the game and am a white mage. I'm so lost. I really want to continue the main story line because it really feels like a Final Fantasy game but I'm stuck at the one where you need to enter the antitower =\ It requires ilevel 180 and I'm only ilvl 142. I did the msq up to here and pretty much every dungeon that I can but I don't remember any of them dropping gear 180+ =\
You should've amassed some tomestones of esoterics by now, which are able to be used to purchase item level 200 gear at a NPC in Idyllshire. And as mentioned above you should get to level 145 (there's a vendor that can knock you up to it) so you can run Neverreap and Fractal Continuum as that'll unlock level 60 roulette, which I mention below.
You can run level 60 roulette , as well as trial roulette once a day to earn bonus tomestones. Additionally, at the same time running these roulettes (albeit at a reduced rate as it's the newest endgame currency) you'll earn lore tomestones which can be used to buy item level 230 gear.
For now your next short term goal be getting to item level 175 which will allow you to do the dungeons introduced in patch 3.1, Saint Mocianne's Arboretum and Pharos Sirius Hard, which will drop item level 185 gear, as well as allowing you to do Alexander: Gordias and Void Ark. Alexander Gordias drops tokens that can be redeemed for item level 190 gear (1 chain for a belt, 1 bolt for an accessory, 2 lens for a head piece, 2 pedals for a foot piece, 2 cranks for gloves, 4 springs for legs and 4 shafts for a chest piece.) which while not the fastest way to gear up you can get some quick accessories and if you advertised that you had a first clear bonus in party finder you'd probably get some people willing to run you through it and teach you it, plus you'll need to clear it eventually to run the next wing of Alexander.
Void Ark drops item level 200 gear, which isn't upgradeable but in addition each time you run Void Ark you get an item which can be used to upgrade the gear bought with the esoterics tomestones.
You can use
http://ffxiv.gamerescape.com/wiki/Guide:Heavensward_Progression_and_Level_Locked_Content as a reference to see if you missed any optional content and while it makes no mention of what's outdated in terms of progression it will point you towards the item levels required for stuff. You can pretty much ignore Bismarck and Ravana Extreme at this point as far as progression goes because their only point is weapons and you can just buy a better weapon with the Esoteric tomestones.