THIS IS VERY LONG.
I'll Just repost something I wrote ages ago with some updates. My original post had quite a bit of focus on not being outleveled for various content, but with the addition of the new Trails feature and the option to reduce XP gains by half, there really shouldn't be any problem with outleveling content. However, that does assume that you will compete the primary zone objective initially, move on, and then return at some later time for more exploration, shard collecting, etc.
Despite the fact that only a few areas are actually required, and that you can play the whole game and get decently leveled without venturing to most of them, many of them DO have story connections to the main plot. Perhaps the only exception might be the Forbidden Oasis, as it is primarily focused on solving the mystery of the shards.
There are also story hints throughout each zone -- primarily codex entries -- that suggest the story order. There is also an "urgency factor" that I mentioned in my original post. By this I am referring to dialogue or other things that make it seem like whatever quest, mission, or objective must be completed immediately. Yes, the game does allow you to complete most things at your own pace, but if following along with the story, and the narrative elements that support it, is your main objective, then I think it would be silly to ignore these cues.
Please keep in mind that all of these are my own suggestions and recommendations. They are based on hundreds of hours of gameplay and what I feel is the best order for everything. That said, a bit of this does come down to personal preference, and that is unique to every player.
To the list...
1 Hinterlands
----- In Hushed Whispers / Champions of the Just - I strongly recommend doing either of these as soon as possible. You can get the required Power Points by setting up camps, closing low-level rifts, recruiting companions, and doing some miscellaneous Haven quests, thereby leaving most of the Hinterlands untouched for later exploration with the full party of your choice (including Dorian + Cole), which is not available until after you arrive at Skyhold.
Note: You MUST complete Templars to the West and Apostates in Witchwood before In Your Heart Shall Burn or else they will auto-complete after that mission and the NPCs will all be gone.
2 Storm Coast
Storm Coast can be done either before OR after arriving at Skyhold. A strong argument for doing it before is the conversation with Leliana about missing Wardens, as well as ambient banter heard in Haven's tavern between two scouts discussing attempts to locate them in that area.
If done before you have the added bonus of Iron Bull recruitment and his subsequent appearance in Haven. There is also a Chargers mission (by talking to Krem) that (I'm pretty sure) you can only do if you recruit him before heading to In Your Heart Shall Burn.
I personally only concern myself with the Hinterlands at the beginning of the game. I'm not too keen on Iron Bull, so I don't find it necessary to recruit him that early. I also feel that there is quite a bit going on between Haven and the Hinterlands, travel time and so forth, and it just feels right to me to leave everything else for after arrival at Skyhold. But this is my own preference, you might want something different.
----- In Your Heart Shall Burn, arrival at Skyhold - Welcome Inquisitor!
3 Fallow Mire
I suggest leaving this for after your arrival at Skyhold and the primary reason is Josephine's dialogue relating to the judgement of Movran the Under. She states, "After you returned from the bogs..." which to me implies a return to Skyhold, bolstered by the fact that Movran was indeed attacking Skyhold.
4 Exalted Plains (civil war, part 1)
Do NOTE that if you leave all of the civil war stuff until after the Winter Palace some of the dialog with NPCs is (can be) different. There is no particular reason for placing this map in this position other than the level requirement. However, you do want to complete it before both Emerald Graves and the Winter Palace mission due to codex entries throughout the area indicating spread of the Freemen. You may also return after attending the ball to talk to some Orlesian officer NPCs and see their reaction once the civil war is officially ended.
5 Forbidden Oasis
This zone can really be done at any time, or never if you don't care for shard collecting. I place this here for personal RP reasons.
6 Crestwood
Due to the "urgency factor" mentioned above, I recommend waiting to talk to Hawke until you are ready to head to Crestwood. Hawke's comments make it seem like they're expecting you to head out very shortly, so hanging around Skyhold, or venturing to some other areas, doesn't really follow along with that. However, this does leave Varric unavailable for any dialogue at his standard location in the great hall, so if that bothers you you might want to go ahead and talk to him anyway.
In Crestwood I also suggest taking Caer Bronach, draining the lake, and closing that rift before leaving the zone, even if you don't do anything else there. Again, the urgency and dire situation of the townsfolk suggest that this is a problem that can't wait.
7 Western Approach
At this point you head to this area to continue along the story path to Here Lies the Abyss. If you follow along with Scout Harding's suggestion about busting the mining operation, you will find a note that mentions Griffon Wing Keep. Because of this, I suggest taking Griffon Wing Keep before leaving the zone, but after you have already done the whole sequence with Erimond.
----- Here Lies the Abyss - While you CAN do either Abyss or Wicked in any order, because of the whole sequence of events relating to speaking with Hawke it makes more sense to me to do Abyss first.
8 Emerald Graves (civil war, part 2)
Resolve the final Freemen of the Dales conflict there before moving on.
----- Wicked Eyes and Wicked Hearts
Afterward, talk to Morrigan in Skyhold's garden at least once before meeting with her in the war room.
9 Emprise du Lion
Be sure to clear out the mine, take Suledin Keep, AND talk to Mistress Poulin again before leaving the area. Because of codex entries found in the mine, complete Red Water after clearing it out.
10 Hissing Wastes
This can be done before or after What Pride Had Wrought. Because of the structure of some of the dialog I recommend before, leaving the final missions to be done in succession.
The downside to this method is that it leaves you hanging around for quite a while, doing nothing, if you want to complete war table missions before finishing the main game. You may use this time to finish your shard collecting, or any other miscellaneous things you'd like to do. Or you can simply just focus on the story and ignore the war table objectives.
----- What Pride Had Wrought
----- The Final Piece
----- Doom Upon All the World
EPILOG - Enjoy the party + optional boy/girlfriend time
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ADDENDUM for DLC
The Descent - I can't give advice relating to The Descent simply because I have yet to play it. However, I will say that the war table objective to start the DLC describes it as helping Orzammar -- a key lyrium supplier to the Inquisition -- deal with a darkspawn incursion. So, despite the large level range of the DLC itself, requiring that you return at various points throughout the game to finish it completely, you can use the lyrium supply angle to justify starting it at any point after you arrive at Skyhold.
Jaws of Hakkon - Because of the very high level range and the plot of this DLC, I do suggest playing it after the main game itself. There is so much stuff going on and the Inquisition, and the Inquisitor, is going to take time away from the end of the world to learn the final resting place for the last Inquisitor? It just seems impractical to me, from a story standpoint, to complete it before the end of the game.There are also some small dialogue differences relating to the defeat of Corypheus (or not) and such depending on when you play this.
Trespasser - Post-game play is required. Also, after the post-credit screen you are thrown back to the game menu with the most recent save being whatever save you made (hard, quick, auto) before the final cinematics started rolling, starting with the final confrontation of Solas.
Now, to me, because Trespasser takes place two years after the events of the main game, there is an argument for leaving both previous DLC to be played post-game. After all, you have to have to do something during those two years, what better than to fight through the deep roads and discover the final resting place of your predecessor?
If this is your chosen path, I do not have a recommendation for whether to play Descent or JoH first other than to say that some of the new focus abilities gained in JoH will help you deal with fights in Descent. You may also use the DLC release order of JoH, Descent, Trespasser.