Going off of your “hoping to stumble on where a quest is”—the main thing you should focus on as a new player is the Main Scenario Quests (MSQ). Patch 4.2 actually added a nifty little MSQ UI element that appears detailing your next MSQ, and, when you click on the element, it will pop up with a map showing you where you need to go to find the quest (all MSQ quests are marked with a meteor-like yellow emblem with an exclamation mark in the middle.

Here is what the UI element looks like (it’s called the “Scenario Guide”):


The small quests at the bottom are usually important quests like class/job quests that you will need to do in order to progress whatever job you have chosen to play as (in your case, the class Arcanist, which actually has 2 job options when you hit level 30–Summoner or Scholar).