Outside, 3 different tentacles spawn. Raid wants to stay generally together.
Constrictor Tentacle - 20k HP, grabs a person and keeps em immoble. First priority to kill, die fast.
Crusher Tentacle - 400 or 600k HP, forgot which. If nobody is in melee range, it channels a -20% dmg done debuff on the entire raid (stacks if multiple crushers up). Every time it gets hit, it gets a stacking buff that increases damage done. Best way is to just ranged them down, don't go in melee. If you get 2 up, the tank offtanks 1, kill other (so only 1 stack of -20% dmg). Second priority to kill.
Corruptor Tentacle - 60k HP I think. Last priority unless you have a good afflic lock (Which is ideal, but not a big deal in 10man). Anyone that can do a lot of damage for 1 global (dots basically) or melee should be on these. They cast some random annoying debuffs including curse of doom, a snare/slow, and a thing that randomly stuns. All are removable/cleansable.
Other things outside:
Random -12% sanity + low dmg.
Occasionally 2 people will get a beam connecting them. If they are too far apart (beam is red), they take dmg + sanity loss. If beam is yellow/gold, you're fine.
Sanity wells are green beam things in the 4 corners (assuming you talked to Freya and got her help).
Inside:
4 (i think) portals spawn when he opens portals. Send 1 healer, and 3 dps (melee first, since they can't kill crushers) inside. When you get inside, there's 1 of 3 illusions. From the time portals open up, you have 60 seconds inside. Each illusion has 2 things going on:
1. "Yellow" (as far as aggro goes) mobs. When you hit one, it turns into an influence tentacle. Influence tentacles have some damage reflect. Have to kill them all.
2. Floating skull things. As long as you look at them, you take dmg + sanity loss. DPS with your back to them, ideally.
Once all influence tentacles die, the illusion shatters and Yogg is stunned (all tentacles are stunned outside...any melee/tanks outside can help DPS crushers during stun). A path opens to the brainroom for people inside. When you target the brain, it's channeling some spell with a 60second total cast time. There is also a portal near the brain. If you don't use the portal before the channel finishes, you get perma-MCed, so make sure you get out with a few seconds to spare. Otherwise, go all out on the brain.
Once you get back out, check sanity, etc. Rinse/repeat. Note portals will be up like ~10sec after you get out (maybe 15-20, idk, it's faster than the p2 start -> first portal pop).
P2 ends when the brain (and Yogg, who's HP is linked to brain) hits 30%.
Ideally, if you want the easiest time in P3, you want 0 crushers up...basically, the last portal you take (when the brain will hit 30%), have DPS inside shatter illusion, then wait at the brain and have it hit 30% as late as possible so the outside crew has as much time as possible to get down crushers while they are stunned.