Ugly Baby - Kaer Morhen
This is the first story quest of Act II and begins immediately after finishing the In Ciri's Footsteps quest (which requires you to complete all story quests in White Orchard, Velen, Skellige and Novigrad). For this quest you will need to travel to Crow's Perch in Velen where you'll find Uma.
Depending on whether or not you complete the Return to Crookback Bog Quest you may be speaking with the Bloody Baron here or his Sergeant. Either way, when you arrive Geralt will demand Uma from whomever is in charge and then ride off into the night with him.
After leaving Crow's Perch there will be some scenes after which you will find yourself in Vizima in an audience with the King. You can respond however you wish to the king - when the audience is finished there will be some more scenes after which you'll regain control of Geralt in Kaer Morhen.
You'll have three tasks to accomplish in Kaer Morhen in order to advance this quest forward. It doesn't matter what order you finish them in so just pick one of them below and get crackin'!
Help Yennefer - Disturbance
All you have to do for this quest is head downstairs and use the potestaquisitor item that Yennefer gave you to locate the disturbance. You'll find the disturbance nearby Vesemir - it's a pair of boxes right next to him. Vesemir will offer to move the boxes for you and then all you have to do is return to Yennefer and speak with her to complete the quest.
Help Lambert - The Final Trial
Speak with Lambert at Kaer Morhen and tell him you're ready to help him out to begin this quest. You'll have to follow Lambert out of Kaer Morhen to the north - be careful when going down the mountain as it's easy to slide off. As you approach the bottom of the mountain you'll encounter some Harpies which you'll need to defeat with Lambert.
At the water's edge you'll find a dock without a boat - after a scene with Lambert your quest will update and send you to the northwest to find where the boat drifted to. Here you'll have to fight some Drowners (pictured above) and a Water Hag. Defeat them then sail the boat to the north.
When you arrive at the cave you'll have some dialogue where you hear what sounds like a kid screaming (it's so not - you can hear the reverb in the voice). Hunting down the sound of the voice will lead you to three Foglets, it's better to just ignore it and head inside of the cave. The path through the cave will be pretty linear - just follow the white dots on your compass if you're lost.
You'll eventually come to a large room with a sleeping cyclops (pictured below) by the name of Old Speartip - you can either sneak by it or fight it, it doesn't really matter. Be sure to grab the treasure chest inside of this room before proceeding.
Outside of the cave you'll find a Place of Power that you'll want to draw from for a free Ability Point after which head up the mountain a little bit more to find three Rock Trolls. You have two options, you can either leave your weapons and armor with the trolls temporarily as you climb the mountain and complete the ritual or fight and kill the trolls. The choice is yours.
Tip: Letting the Rock Trolls live gives you credit towards helping monsters in the Contract: Skellige's Most Wanted Quest.
Help Eskel - To Bait A Forktail...
Just out front of Kaer Morhen you'll find foot prints that you will have to follow for the start of this quest. They will lead you to Eskel's camp which is to the south of Kaer Morhen (pictured above). Here you will need to investigate Eskel's supplies near the campfire (which will have Superior Draconid Oil in them if you don't already have it) and the foot/hoof prints near the camp.
After inspecting everything your quest will update and send you south (following the foot prints) to find Eskel. He'll have the goat tied to a stump and upon approaching you'll get a scene with him. During the scene you'll be interrupted by the Forktail, after getting it low in health it'll flee from you into the mountains - follow its blood trail to a nearby cave.
Inside of this cave you'll find not only the Forktail but also the Diagram: Enhanced Wolven Gauntlets. When you're done exit the cave and Eskel will challenge you to a Horse Race - if you beat him he'll give you a Crafting Diagram you don't have as well as a piece of armor.
Return to Kaer Morhen
Once you've completed these three tasks you'll want to return to Kaer Morhen and speak with all of the Witchers in the main room. There will be a bunch of dialogue and some scenes. If you were romantically involved with Yennefer in the the past then during these scenes you'll be given the opportunity to slip away with just Yennefer and sleep with her again.
Note: During the scenes you'll have the option of playing Gwent against Lambert. Beating him will reward you with the Triss Merigold card. There will be another opportunity to get this card during the Gwent: Old Pals quest chain.
You'll control Geralt during some of this but it's all very linear and easy to figure out, when it comes time to brew The Decoctions of the Grasses you'll need to open your Alchemy tab and do it that way. The quest will end once all of the dialogue and scenes are finished.
Completion of this quest will add many followup quests to your log, the main one being The Isle of Mists. However, Brothers In Arms: Novigrad, Brothers In Arms: Skellige, Brothers In Arms: Velen and Brothers In Arms: Nilfgaard will all be added to your log as well.