Welcome to FFXIV!

As far as gear goes for your early classes, you can indeed buy the gear from NPCs pretty cheaply. As Gumbercules said, your physical DPS classes (and, early on, Tanks, though they'll start getting their own sets around level 15) will share gear for Disciples of War, whereas the others (magical DPS and Healers) will share gear for Disciples of Magic, and your class quests will provide you with upgrades to your weapon (as well as one piece of armor that you can choose from).

It's worth noting that the gear awarded from completing the Hall of the Novice exercises available starting at level 15 (one Tank set, one DoW set, and one DoM set which you can acquire by completing the exercises as those disciplines) can be used up to around level 30, too, which saves you the hassle of getting your hands on a few sets of gear. You'll also get a ring that increases your exp. gain below level 30 when you complete the final exercise for the first time.