Something worth noting when people say "level 15", level 35", etc: all story quests have their own level, separate to your character's level. This is listed on the "title cards" you see when you accept it.

You're not able to undertake quests that are higher than your character's level, but you can level your character well above the current quest level. Technically you could grind all the way to Lv70 on FATEs while still at story level 5, without unlocking any of the features you're supposed to have at that point.

A side effect is that a lot of sidequests become available (as they appear based on your actual level) long before you're actually intended to do them. So you might feel like you need to go running off to distant corners of the map because you can see quest icons, when actually you don't and shouldn't.

If you want to do sidequests as you go, only pick them up from the location where you have currently been sent by the MSQ storyline. But it's fine to leave them and come back later.



From memory, major things that will unlock at certain levels of the MSQ:
Lv10: Guildhests - your first access to multiplayer content. Short-form battles with a specific objective or gimmick.
Lv15: Access to other cities. A sidequest in Ul'dah unlocks the "Gold Saucer" minigame casino.
Lv15-17: Your first three dungeons. Somewhere around here (I believe after being told about the first dungeon) there is a "blue icon" sidequest that will unlock the Challenge Log, a weekly list of activities that will give you lots of bonus gil and EXP for completing objectives.
Lv20: Join a GC, start earning their specific currency, and gain access to mounts.

You'll also want to keep up with your quests from the Thaumaturges' Guild (and any others you join) - finishing the Lv10 quest allows you to join other guilds, extra skills are unlocked by quests at Lv15 and Lv30, and then you progress to the 'job' of Black Mage.



Also, a couple of things that do unlock by your level rather than MSQ level:

At Lv15 you gain access to the "Hall of the Novice" which will get you some nice gear and a ring that boosts EXP gain.

Also at Lv15 you can get access to using dyes and glamours, and start making your gear look nice! Go to Vesper Bay in Western Thanalan and do the blue sidequest there.

At Lv30, as long as you've unlocked mounts, there's a sidequest at Camp Tranquil in the South Shroud that lets you summon a chocobo companion in battle. This is a good damage boost for solo play and they can also cast healing spells.