I don't want to spoil the fun of discovery too much for you, so I won't get into really specific details, but I'll mention just a few things to start, otherwise I'll be typing for hours. XD
What I love about the Souls series, probably the most common thing we've discussed and part of what drew me into DDrop as well when the kickstarter was going was the rich world-building, I can't express how important that is and how much appreciation it garners from myself and many fans of the series, and going beyond general info about a world, the stories you can tell with characters, locations and even weapons and spells can lead to some great discoveries by the player.
I'll use an example from Dark Souls II: There's a boss in one of the DLCs who ties into the lore of the main game, and it just so happens that when you fight him, if you wear the helmet piece of his rival who is also present as a boss in the main game, he skips his initial fighting tactics dual-wielding two swords and turns his ultra greatsword into a flaming deathstick.
He does this anyways partway through the fight, but the intention is that his reaction to seeing you donning the helmet of the man he held a bitter rivalry with enrages him (either because it means you defeated his rival and took the opportunity of revenge from him, or just because the reminder angers him in general for what he lost over it) and makes him take you on at full intensity from the very beginning of the fight.
That's not a super common thing in the series, but I love that the lore so directly impacts the gameplay in that instance. There's a lot of great examples of storytelling throughout the entire series, and that's just one of many favorites of mine.
Dark Souls itself has a small variant on the intro cutscene of a boss if you complete the Artorias of the Abyss DLC beforehand, it's such a tiny change, but it changes the context and the reason the boss is fighting you, and makes it all the more tragic.
I'll say no more though, play how you want this first time through, then maybe come back to it at some point. I know your time is limited, but each entry in the Souls series really does lend well to repeated playthroughs. Also note that your first playthrough will probably take the longest, so return trips will feel smoother since you'll be familiar with the mechanics at that point.
If you do want to find more straight-up lore once you've played through the game, I recommend EpicNameBro's older videos, as he used to do some really good stuff on DkS1. VaatiVidya does a mix of DkS1, DkSII and Bloodborne lore videos which are quite nicely done and develop some narratives which may initially seem vague into really robust stories about different figures within the world.