Delphine will tell you to find a guy in Riften, to quickly get to another city you haven't discovered you can head to Whiterun (or any other hold) Stables and ask for a ride for pocketchange.
Inside the Hold, your target will tell you to do something for him in exchange for Esbern's location.
Wait till it's morning, then talk to Brynjolf again, then head to this stand and open the Sliding Door and the lockbox within. If your pickpocket is high enough, you can potentially not fail this quest, but I'm a fat lazy orc and I got caught.
If you weren't caught, put the ring in your target's inventory via reverse-pickpocketing, however, if you were; pay the bounty or die. The bouny is about 6 gold.
Brynjolf will be pissed if you fail but honestly who cares, more importantly he'll give you the location to Esbern, in the Rataways. Follow the quest tracker, killing junkies and no-lives in the Rataways till you hit the Thalmor who are armed and dangerous.
Keep heading forward, you'll eventually find yourself in the Thieves Guild meeting hall, talk to the bartender and bribe him to get the location of Esbern.
After some seriously tedious lockpicking doors and Thalmor, Esbern will be behind one hell of a locked door, luckily he'll open it for you, just tell him something about the 30th of Frostfall, then tell him you're the Dragonborn. Esbern will then ask you to lead him to Delphine.
Immediately after escorting Esbern away, the Thalmor will attack. Kill them and head back to Riverwood.
At Riverwood, the two will reunite and smooch (hot) and then proceed down to the secret room for a boring conversation.
They'll then decide to head to Karthspire, either you can follow them or make it there yourself.
It's near the province of Morthal. I recommend you follow them if you don't want to carriage to Morthal.
Head to Karthspire, where you, Esbern, Delphine, and probably Lydia or someone else will be assaulted by a bunch of Forsworn and a goddamn dragon.
Once everybody's dead, head inside Karthspire with your party and slaughter the Forsworn in there.
You'll see a little puzzle with three pillars you need to turn. Turn them all to the little heart-shape with the arrow to lower the bridge and proceed.
At the fire-traps, Whirlwind Sprint to the machine VERY QUICKLY and pull the chain or you'll be literally blown apart by fireballs.
Then at the gate, Esbern will make a dialogue, blah blah blah, then you need to press E or E-quivalent (hahaha) on the circle in the middle, where the Dragonborn will cut himself because nobody understands him, thus opening the gate.
At the Wall, Esbern will tell you to go to the Greybeards to learn a shout. Exit Karthspire and fast-travel to Hrothgar.
At Hrothgar, Arngeir will say no, but then say yes (so indecisive) then tell you to head outside and he'll teach you a shout to get you to the peak of the mountain.
Once you learn the Clear Skies shout from Arngeir, equip it and shout at the fog to clear it. Head up all the way to the Throat of the World, wary of Ice Wraiths that pack a punch.