There are 3 major questlines in Skyrim Special Edition: the main quest, the Dawnguard DLC, and the Dragonborn DLC. These have been expanded upon to allow for player choice and a true RPG experience. Additionally they have been immersively delayed as these quests are not early game content. This guide will detail how to start them.
To start the main quest you have several options. First of, while not recommended, in the starting room you can select the dragonborn option which automatically start the main quest. However if you pick any other option, Helgen will not be destroyed and you'll be free to explore Skyrim. When you are ready to start the main quest head the inn in Helgen and rent a room (there will be an option to start the main quest so if you just want to rent a room you can do that as well). From there, when you wake up a familiar scene you take place, but you will have an alternate perspective.
From here there are several key changes to the main quest. First of all, you can now Storm the Thalmor Embassy which allows you to skip the “party” and go straight to the Thalmor Embassy during “Diplomatic Immunity”.
Defeat the Dragon Cult expands the later half of the main quest, giving the Dragonborn a reason to avoid facing Alduin until all the Dragon Priests are dead.
The Blades have also been expanded. Paarthurnax - Quest Expansion provides an option where you no longer need to kill Paarthurnax whilst giving both the Blades and Paarthurnax good reasons for their stances, and you, Dragonborn, the final choice. Dragon Hunting has also been improved with improved rewards from the Blades and exclusive alchemy ingredients from dragons for a worthwhile hunting. Also checkout Redeeming Fultheim - A Blades Quest Addon.
The main quest was very linear and rather short in the base game but LoreRim has expanded upon to provide more choices which can cater to your playstyle and lengthens it with the added Dragon Cult.
To start the Dawnguard DLC questline, first the quick solution is to just head straight to Dayspring Canyon where the quest will auto start. However, vampires are dangerous enemies and it is not for the low levelled. In vanilla Skyrim, the Hall of the Vigilant is destroyed when you reach a certain level where you will be approached by recruiter for the dawnguard. In LoreRim this is delayed with Sensible Quest Prerequisites which requires the quest "Laid to Rest" to be completed. You can start this quest by entering the burned down house in Morthal.
Dawnguard came with some additional quests involving curing vampirism, which has been expanded for both the Dragonborn and Serana. Seeking The Cure repurposes “Rising At Dawn” into a non-repeatable, story-focused quest where Falion’s ritual to cure your vampirism fails. Serana Cure Quest Plus adds more to Serana’s cure quest.
To start the Dragonborn DLC questline, after completing "The Way of the Voice" you will be attacked by Miraak's cultists.
Same as Alduin, Miraak now has a cult of dragon priest which provide him buffs. Kill Miraak's dragon priests to break his influence and weaken him... or kill his challenger to strengthen him.