It really depends on your interests and goals. What do you want out of the game? Only by knowing that, can we really help you. Common examples are...
  • Story/lore person.
  • Raiding/difficult content.
  • Social/to interact and make friends.
  • Roleplaying.
  • Collecting things (achievements, mounts, minions, or whatever else).
  • Pvp.
Everyone plays MMORPGs for different reasons, or a combination. If we tried to explain everything about the game, we would be here a long time writing posts. So knowing what you actually want out of the game will help us narrow down our suggestions for you.

The main thing with the quests is that the fireball quest is the most important one. It has an endgoal, which is unlocking all the areas and its associated content, although it sometimes gives other things such as experience, mounts and minions.

The blue quests unlock content or game features, but are technically optional.

To participate in raids, you would just queue for them using Duty Finder. Although it depends which type of raids.
  • Binding Coils were meant to be the hard raids for A Realm Reborn, so they are not supplied by a roulette.
  • Anything labeled (Extreme), Minstrel's Ballad, (Unreal), (Savage) or (Ultimate) is meant to be difficult content, and isn't supplied by a roulette, so the queues could take days or never succeed. Extreme and Minstrel's Ballad are supplied by Mentor Roulette, but only a relatively small portion of players have access to that, so the queue could still take hours or days.
  • Alliance Raids (24-person raids) are casual content and you can queue for any of them and it will be fine.
  • Starting in Heavensward, the 8-person raids have an easy story version and a (Savage) version. So if it does not say (Savage), just queue for it and you will eventually get into it and it'll be easy.
  • People don't do old "hard" raids in their challenging form very much. They just destroy them easily in their higher-level gear, or focus on the latest ones at max level. But there are small communities who do old raids with settings that make them difficult again, for fun/the experience of doing it.
  • "Hard" raids are typically recruited for either in Party Finder, in discord PUG servers, or by forming statics through discord, reddit or websites such as https://www.reddit.com/r/FFXIVRECRUITMENT/ or https://www.xivrecruit.com/