There will be times where you are unable to progress in the main story because they are gated by Adventure Rank level. You get Adventure Rank experience when you do various things in the game. To put it simply, play the game and explore, open treasure chests, collect anemoculus/geoculus to offer to the shrine and do dungeons when available.
However, there are fast(er) ways to level up.
Daily missions from AR 12.
Story side missions.
Domains or Ley lines, though they cost 20 resins to clear each.
Why Increase Adventure Rank?
Increasing Adventure Rank is important since it lets you progress in the main story. You also get many rewards from the Adventurers' Guild each level for free. Certain features are unlocked when you increase your Adventure Rank as well, like multiplayer, expeditions, and dailies. Last important thing is that it can increase your World Level once every 5 levels after you reach AR 20 and do an ascension quest. Increasing your world level will increase the rewards that you get whenever you get items.
AR 12 - Dailies
Upon reaching Adventure Rank 12, you will unlock the daily missions by talking to the Adventurers' Guild. Once you do this, you will get 4 tasks each day. Completing each task will give some adventurer exp. Completing all 4 of them, will give you an additional 500 exp. You have to talk to the adventurers' guild to receive them after completing 4 tasks.
AR 14 - Dispatch Expedition
At Adventure Rank 14, talk to the Adventurers' Guild again to unlock the Expedition feature. It lets you dispatch your character(s) out on an expedition for a certain amount of hours for some items, but early on you can only dispatch 2 units at a time. When they are out on an expedition, you won't be able to use them unless you cancel the expedition, or it's completed. Certain characters have the passive ability to reduce expedition time by a certain amount like Fischl, or Bennett.
AR 15 - Ascension
AR 15 is when you get the item needed to ascend your main character to increase their max level. Talk to the Adventurers' Guild to receive this item.
AR 16 - Multiplayer/Co-op and Domains
Adventure Rank 16 is when you can start playing the game online multiplayer, or co-op. Multiplayer works like a host and join session, not like an open world MMO where you get to see hundreds of players walking around. In order to join, you need to be at the same world level with each other. The shortcut for the multiplayer is F2 on pc.
The game will show you that you can start doing domains. You can view available domains by going to your adventure handbook, and then the domains tab. Domains are dungeons that changes daily and give various rewards like items needed to ascend your characters or weapons.
AR 20 - Battle Pass and World Level
At AR 20, you will unlock the Battle Pass. There is a free version that everyone will automatically get, and also a paid one and lasts for about a month.
Complete certain tasks to get battle pass exp, and earn rewards whenever you level up.
Increasing your World Level will better your rewards that you get from the world map, so it is recommended to try and get AR 20 before you fully explore the map and take treasures like luxurious chests.
AR 25 - City Reputation
At AR 25, you will unlock the city reputation system. You will be awarded for your exploration's progress and get new weekly tasks and bounties. Leveling up your city's reputation will reward you with new items that can help your exploration.