Frankly, you'll pick up a lot the background just doing the zone/faction quests. The lore and background is told through those quests. I'm trying to think of how to recommend specifics without just following the story and I can't, because it just won't make much sense.


Argonians and Dunmer
1. Bal Foyen gives some backstory, being a former Dunmer territory ceded to their slaves.

2. Part of Stonefalls' main quest and a number of side quests deal with the fraught relationship between the Argonians and Dunmer (and Nords too) as exposed by the stress of the Covenant invasion.

3. Shadowfen is the only base game Argonian area, so there is more background here.

4. For an alternate view on the situation, in Grahtwood there's another Argonian tribe hoping to join the Dominion.

5. On Vvardenfell, in the Telvanni territory, you can help an Argonian rise through the ranks of House Telvanni. The rest of Vvardenfell itself has some extra information, but frankly you won't learn anything you didnt learn from Stonefalls.

6. Murkmire is another Argonian area, so again, you'll get more Argonian viewpoints.


Dominion Bosmer and Khajiit
If you want the stories to make sense, please, please, please do the quests in order. The Dominion is an extremely linear questline and things will not make sense if you do zones out of order!

Bosmer:
1. Khenarthi's Roost introduces two major fixtures in Bosmer society: the Silvenar and the Green Lady.

2. Grahtwood, then Greenshade, then Malabal Tor in that order are the Bosmer zones. Reaper's March, the next zone in the faction has some Bosmer areas. Please note that a major plot point in Grahtwood and Greenshade won't make much sense unless you've done Auridon before them.

3. City of Ash 1 and 2 deal with the fate of a bosmer settlement.

4. Coldharbor has a Bosmer settlement.


Khajiit:
1. Khenarthi's Roost is a Khajiit island in the middle of a treaty dispute.

2. Reaper's March is the only base game full zone of Khajiit land and has a ton of info on their culture and politics. Its also the final chapter of the Dominion questline and makes more sense as such.

3. Elsweyr is a whole Chapter about the Khajiit.


Covenant and Orsimer
If you want to know more about the Orc culture, there is no substitute for the Orsinium DLC. Its an incredibly detailed exploration of the new Orcish kingdom.


If you only want one or two recommendations, I'd say:

Argonians/Dunmer: Stonefalls, Shadowfen, or Murkmire

Bosmer: Malabal Tor. No it won't make complete sense, but its a more self-sufficient story than the other two.

Khajiit: Elsweyr

Orsimer: Wrothgar

OP: you sound like me, which is unfortunate for you. I'd recommend the AD storyline, as it's the most cohesive and designed around a real tour of the aldmeri Dominion. The EP and DC zones are frankly very scattershot, random, and IMHO kinda dull.

Even though Im no fan of the high elves, it's simply the only faction that hangs together with an engaging story. EP especially I found it's just a never ending stream of random NPCs without much context having random problems that don't have much to do with anything.