There should be charts of stories order somewhere on the internet, first of all.
If you skip side quests you may miss out on some recurring characters or (in case of chapters and dlc) have less people at the usual celebratory gathering at the end of the main quest line of the zone.
But mainly by skipping stories you miss out on, well, stories.
Main questline: The hooded figure gives you the quest to start you off...these are the harborage quests that reward 1 skill point each quest for a total of 10 skill points..Skill points are very helpful in the early stages especially to get yoir skills you need to build your character.
Then i go to mages guild and fighters guild and i join and there are the crafting trainers so i pick up crafting quest ....i do my crafting dailies ever day pick em up from writ boards and they reward quite a bit of gold which helps for horse feeding and bag spaces
Then i do the psyjiic questlines which also awards skill points and gold and by dojng it i open up wayshrines along the way and collect lore books to get my mages guild rep.
At level 10 i start my dungeons....all this levels me up quite fast.,
Well, doing quests matter if you really want to follow the story, if you are doing them only for the skill point or reward you can just do the Main and Guild Questlines, I suppose.
And if you want to learn ESO storyline in the right order, I suggest you follow a very dedicated guide, because ZOS messed things up, not giving a common starting quest, instead anyone can start at any point (minus the forced tutorial based on the last Chapter you have), adding prologue quests to various chapter/DLC that you can do anytime and not in the "right" moment (some of these also break a couple of Main questlines if active in the journal) resulting in a non-chronological-sense
Do the Main Quest (leads to and unlocks Coldharbour), then:
~ instead of beginning Cadwell's Silver and Gold (kind of a confusing list), just go to each Zone and do the Zone Quests which will net you MANY Skill Points overall, and when finished you will be only a few incidental quests away from Cadwell's Silver and Gold completion.
Start Zone Quests with the small Starter Areas (btw, start Stros M'Kai before Betnikh) and you will see some of the Lore and really get a feel for the game in a general sense.
And, make sure to do the 30 quests needed for the Dragonguard unlocks since you will get access to a daily crate with goodies inside.
Make sure to do the Deshaan Zone quest that leads to and unlocks Clockwork City. Do the Psijic quests to unlock Artaeum and the Mage Guild quest line to unlock Eyevea -- an interesting "hidden" island (click on the "Ship Drawing" in the ocean north of Summerset to see the "hidden" island.
Also, be smart and do all your craft skills to 50 and begin whatever crafting Research you want to explore. Do eight out of nine Traits (Nirnhorned optional, Triune for jewelry optional since both are rare and the trait materials somewhat difficult to get.)
(Get all Skyshards as well -- LOTS of Skill Points from them.)
~ PS NA ~ ALDMERI DOMINION ~
IVY GOLDBLADE, WOOD ELF NIGHTBLADE, Former Empress
Honestly, I'm only doing this on my main character. Started a couple of month's ago and I'm around CP 540 now. I've done most of the zone main quests, but still plenty left. I've made a second character, but I only did main quests on that one unless I need more skyshards because it can take up so much time.
Do you like questing/lore/stories? Then you should complete everything. I like to min-max my character in endgame, but I do like to hunt achievements so I'll eventually complete everything ^^