If you are playing for the first time, do play at least through the 2.0 scenario before delving into the lore. The most basic need-to-know stuff is covered there and all the lore that comes afterwards builds off of it.

That said, here's a couple of very helpful references that I've found for after you finish 2.0 and want to go through the lore with a fine-toothed comb.

Garland Tools - This is an amazing resource. It has all the Journal Entries, all the unvoiced dialogue and most of the quest-dependent NPC dialogue of every single quest in the game. This lets you browse through the vast majority of the information given in quests without needing to have the game open in front of you. The only thing it doesn't have is the voiced dialogue since the sub-titles for that are stored differently in the game files, but you can find that dialogue in-game in the Unending Journey (in an inn room). Just beware of what you're clicking on as there is spoilers for everything in the game in there... This is my go-to resource for figuring out where/when in the game various pieces of lore comes from.

This site also has a ton of other useful information and functions in it that help with normally playing the game. If you only have time to look at one resource, it should be this one; everything in this is from the game itself, so everything in here is canon by default.

Encyclopedia Eorzea Vol. I (and soon to be Vol. II) - Otherwise known as the Lore Book. If you are a lore nerd, this is invaluable. It's written from the PoV of scholars living in Eorzea and contains so much information in it. A lot of information that is hinted at in the game is explained here in scholastic detail. And much of the information in here never made it into the game in the first place since there's never been a natural place for it to be included. You'll find everything in here from a detailed cosmology to a bestiary to detailed descriptions of all the places in the zones to a history of the Sixth Astral Era. Volume I runs from the start of history to 3.4. Volume II will coming out this November and most likely have information from 3.5 to 4.4. In a nutshell, this is practically the unofficial FFXIV Dungeon Master Guidebook. Spoilers naturally abound in this, so play most of the MSQ before reading it.

1.0 Resources

Okay, so technically the game does explain what roughly happened in 1.0 to the player, but that still glosses over a lot of character interaction that goes a long way in establishing why a lot of the main characters do what they do. Again, I wouldn't look into this until playing at least until the end of the 2.0 MSQ. Just keep in mind that some of the nitty-gritty lore in here was retconed when FFXIV was relaunched, but the broad strokes of it are correct.

The Rise and Fall of the White Raven - This is probably the best text summary of 1.0 and the events leading up to 1.0 on the internet. It's about as close to 1.0 canon as you can get without watching the 1.0 cut-scenes yourself. Many thanks to Anonymoose for collecting all that data and putting it in writing!

Incidentally, the rest of the GamerEscape FFXIV wiki is a good source of basic info about characters, events, locations, etc. Most of the information there is from the game or the Lore Book and there's enough in-game references in there to know where to start looking for more information. There's also quite a few other lore articles and interviews on this site, but most of those are centered less on general lore and more on specialized lore. So I'd take a look at those once you've finished the bulk of the MSQ.

1.0 Cut-scenes - If you do want to watch the 1.0 cut-scenes, this would be the playlist to use. It's got all the MSQ cut-scenes from 1.0 to 1.23 and even includes the 1.0 Hildibrand cut-scenes! This play-through starts in Ul'dah. Fortunately, you can find the other starting city-state 1.0 cutscenes on YouTube. Here's the 1.0 Gridania cut-scenes and here's the 1.0 Limsa Lominsa cut-scenes. You can find other 1.0 cut-scenes on YouTube, but the MSQ ones are the ones that impact 2.0 the most.

One site you'd be better off not going to for FFXIV lore is the Final Fantasy wikia. There's a lot of unconfirmed data on there and some things that are flat out wrong. Let's just say it doesn't have the best quality control and leave it at that.