Ways to get equipment:
- Many quests will reward you with suitable pieces equipment.
- There are NPC vendors in the main cities as well as other outposts around Eorzea. The main market area of Ul'dah is known as Saphire Avenue Exchange found in the eastern side of the Steps of Thal (near the Gate of Thal). There are multiple vendors there, some specializing in weapons, others in armour, others in crafting supplies, etc. Check out what each of them has. For gear, only the lower-level stuff (which of course is what you want now anyway) is found in the markets of the three main city-states. Once you get to wanting higher-level gear, you'd need to find merchants in level-appropriate areas. Oh, and if a merchant offering gear asks if you're looking for "Battle Gear" or "Field Gear", the difference is that Battle Gear is the sort with stats for combat classes like Gladiator or Thaumaturge, whereas Field Gear is for crafting or gathering classes like Weaver or Miner.
- Another thing you'll find in the main market area is a Market Board. This is where items are posted for sale from other players. You can at times find better gear here than what the NPC vendors will offer, since players who've taken up crafting can produce either "Normal Quality" or "High Quality" results. (What the NPC vendors sell is equivalent to a crafter's Normal Quality products.) High Quality (HQ) items are identified with a sparkly background to the item icon, or by a special icon next to their name that looks like two crescents conjoined at the bottom
When you find the same item on both the Market Board and from NPC vendors, the Market Board has widely varying costs, according to what the selling player chose. Sometimes you can get great deals there, buying gear for a lot less than the NPC vendors would sell it, and sometimes it's the other way around, with the MB listing it for far more than NPCs. (NPCs, however, have constant prices, so if you find it at a vendor, the only price checking needed is between them and the Market Board. There won't be any other vendors offering it at different prices.) You can buy stuff off the Market Board at any time, but won't be able to post your own items for sale there until you unlock Retainers from a level 17 Main Scenario quest.
- Once you get a few more levels in, you'll start unlocking dungeons, and their treasure chests can drop items better than you'll find elsewhere. There is Aetherial gear (identified by a pink background on the icon) which has stats somewhat better than regular vendor gear, equivalent to the High Quality results from crafting, and there is special (dungeon exclusive in the early and mid levels) gear with a green background, which is better yet. (Once you get near endgame, there are more tiers, with blue or purple backgrounds, but don't worry about that yet.)
- You can also take up crafting yourself if you're interested, and make your own gear. First you'd need to reach level 10 on your starting class and take the level 10 quest from your class trainer. That unlocks the ability to add other classes as well, both other battle classes as well as crafting and gathering classes. Then you go to the class guild of the class(es) you want to add and take a simple quest from the Receptionist at that guild, which will unlock that particular class for you. So for instance, if you wanted to become a Jeweler, then after your level 10 Gladiator quest, you'd take a quest from the Jeweler Guild and become a level 1 Jeweler.
You alternate between your unlocked classes by equipping that class's main hand weapon or tool. This game is very flexible in multi-classing opportunities, as you can unlock and level as many of the classes and jobs as you want. (Many people do all of them.) Each of those classes needs to be leveled up separately, however, so your level in one class won't be your level in another.
Anyway, getting back to the topic of crafting, once you unlock crafting classes, you'll get access to a Crafting Log, which has a list of recipes you can make. Your class quests in the crafting guilds will teach you how to make stuff. It's pretty simple to start off with, getting a bit more complex as you go up in levels and gain access to more skills that give you a variety of ways to improve your chances at High Quality results.
For other things to know about, some other people have already mentioned that you want to follow the Main Scenario quests (the ones with the flaming circle around the quest icon). The other important set of quests come from your class guild. There's one quest there each 5 levels (and that goes for each of the classes you unlock once you start advancing more of them). Past level 30, the class quests will be replaced by job quests which work much the same, coming each 5 levels, with the main difference being who you get the quest from. (The level 30 quest unlocking a job will lead you from your class guild to wherever it is you need to go from then on, introducing you to your job trainer.)