You can do pretty much all of the content without interacting with other people. Obviously, there are instances that you have to complete to progress through questlines, but there is a LFR (called "Duty Finder") that automatically places you in a group with other people, and the instances are really easy. FFXIV is more of a JRPG with some MMO-lite elements tacked on (based on WoW) then a MMO.
The bulk of the game's content is centered around the main story questline, or MSQ. It starts from the moment you first log in and continues through every patch. There is a tracker in the top left corner of your screen displaying which MSQ quest you are on.
You can switch jobs (classes) at any time, but the MSQ gives you slightly less then the amount of exp required to reach the level cap of that expansion (meaning you will probably have to do some sidequests or queue for some dungeons to get to the level cap of that expansion and continue the MSQ, which is level gated). As such, for your first playthrough, it is recommended that you stick with a single job. Any job can complete all content, even healers. However, tanks and healers get near instant queues for instanced content. If you don't want to wait 10-30 minutes every time you queue for an instance, consider going with a tank or healer job.
Due to the length of the MSQ, some parts of the story had to be split off into their own optional questlines for the sake of pacing, but they are still relevant to the MSQ story and should be completed. The order in which you should do those questlines is:
The FFXIV story order:
- 2.1-2.55 ARR MSQ (do the Crystal Tower storyline during this, only takes a couple hours)
- 3.1-3.55 HW MSQ (do the Warring Triad and Alexander storylines during this. They each take only a few hours. After completing the 3.3 MSQ, talk to an NPC in Fortemps Manor to pick up the quest "The Paths We Walk". That questline only takes about an hour.)
- 4.1-4.55 SB MSQ (do the Omega and the Return to Ivalice storylines during this. This will take some time.
- 5.1-5.2 ShB MSQ (do the Eden and the Save the Queen: Blades of Gunnhildr storylines. If you're willing to level up alt jobs (one tank, one healer, one physical DPS, one caster), you can can do the Rolequests storylines, which are short.
The FFXIV story is probably going to end in 6.0 or 6.3. We'll get more content after that, but that will be the end of the saga that began in 2.0.