How exactly are you expected to upgrade your armor and weapons throughout the game? I've sort of been able to coast so far because I had some lvl 30 armor gifted to me when I started playing, but I'll probably need some better stuff soon. Are you supposed to just buy gear from shops whenever you level up? Or are there quests specifically for getting armor/weapons?
There are a bunch of different ways. You can buy gear from vendors, but HQ crafted gear or dungeon drop armor is generally better. I leveled Armorsmith alongside Gladiator, so I never wanted for gear. However, dungeon dropped armor lasts for a good length of time. A level 30 set of dungeon armor is slightly better than a level 35 set of vendor-bought armor. If you get lucky, you can keep up on equipment pretty decently with dungeon drops.
Other good, if inconsistent, sources for armor and weapons are the Grand Company vendors. They don't give full sets all the time, but you can get some decent armor, weapons, and accessories at a wide range of levels for 1-50 content. If you are playing a mage or other lighter-armored class, the level 20 GC uniform is pretty decent gear til you reach level 30 or so.
Guildleves can also be a source of armor. They can give some NQ armor as rewards, but you also have a chance to get Aetherial armor from the random chests.
You can get a full set of level 15 armor just from doing the training hall quests in Western La Noscea.
Importantly, you get a great set of armor from your Job questline at level 45 and 50. That is what served as the original entry-level endgame gear at launch, when paired with the level 50 GC weapon.
As other people have said though, the main scenario quests give plenty of armor to gear your first class. It doesn't hurt to grab some dungeon drops for any alternate jobs you want to level in the future though. The level 15 dungeon armor and level 30 Brayflox armor lasted me across a lot of jobs.