I've tried to put easily obtainable pets into my combinations so everyone can make a team - but if there are any that are hard to get then I've got some suggestions from other people's teams.
The Team I used to beat Yu'la
Gregarious Grell: Punch, Phase Shift, Cauterise
If you don't have a Grell (which is a TCG Card pet,but you can get it on the AH) then another good option from Liopleurodon is
Feral Vermling: Crush, Sticky Goo, Backflip
Hopling: Crush, Sticky Goo, Backflip
Anubisath Idol: Crush, Stoneskin, Deflection
The Team I used to beat Chi-Chi
Unborn Val'kyr: Shadow Shock, Curse of Doom, Haunt or Unholy Ascension
Rapana Whelk: Ooze Touch, Shell Shield, Dive
Silkbead Snail: Ooze Touch, Shell Shield, Dive
I had a terrible time with Chi-Chi. I tried all sorts of combinations of dragon and magic pets to no avail. However Luxy came up with a solution that she'd read somewhere that I thought was brilliant and surprising! She used an Unborn Val'kyr with 2 snails - and like her, I said "Snails? Really?" The difference is that I use my Rapana Whelk A LOT, whereas she never uses hers at all, so I'm not surprised that snails can be awesome. Haunt is a really cool ability, it looks like you're dead but you're doing damage whilst doing so and that gave plenty of time for the Snails to kill Chi-Chi with their Ooze touch. Chi-Chi doesn't heal, so that helps and its dodge attack (Ethereal) often comes at a time you want to use a big attack, so that is pretty annoying.
If you haven't got an UnbornVal'kyr yet, then a reasonable substitute is
Ghostly Skull: Death Coil, Spectral Strike, Unholy Ascension
The Teams I used to beat Zao
Dragon Kite or Wild Golden Hatchling: Tail Swipe, Call Lightning, Lift-Off
Landro's Lil XT: Zap, Heartbroken, XE-321 Bombot
Netherspace Abyssal: Immolation, Explode, Meteor Strike
The second team is a bettter combination, I think, and more reliable. Zao has Wish, which heals for a craptonne but it requires two turns. If you can do the big damage quickly then you have time to get him down. I used the water spirit's geyser then whirlpool and then cast one water jet before I died. Then Alpine Foxling came out and did Howl and so Zao took double damage from Geyser and Whirlpool and I finished it off with 2 bites. Phew! Didn't even need the third pet!
The Team I used to beat Xu-Fu (also used for Lil Oondasta)
Tranquil Mechanical Yeti: Metal Fist, Call Lightning, Ion Cannon
Clockwork Gnome: Metal Fist, Blitz, Build Turret
Menagerie Custodian: Zap, Shock and Awe, Ion Cannon
Xu-Fu's Feed does a big heal, and when he turns it to moonlight, the heals are bigger, and Spirit Claws hits harder. He usually casts moonlights first, so I cast Call Lightning to cancel that. When my Yeti is only going to get one more hit in, I use Ion Cannon. Then out comes Clockwork Gnome building Turrets to add that little extra damage from the extra hits in Lightning and if I'm lucky, I can get 2 turrets built if Xu-Fu misses one of his hits. But most of the time, I have one, and I use metal fist until my Gnome dies. This is where it gets into a bit of luck. If Shock and Awe misses, Xu-Fu might get a heal off and I may not have enough health to live through to get another zap and an Ion Cannon off. If the Ion Cannon doesn't finish off Xu-Fu (and by this time, the Call Lightning has worn off) then it's the end of the game and I have to reset.
One combination that I've seen someone else use and want to try is
Zandalari Anklerender: Hunting Party, Primal Cry, Black Claw
Zandalari Kneebiter: Hunting Party, Black Claw, Leap
Zandalari Footslasher: Hunting Party, Leap, Exposed wounds
I tried this and it failed! I tried multiple times, but I couldn't figure it out, because each thing would get 2 shot! However, I finally figured out HOW to make this combination work. First you need to Primal Cry (so that your raptors would always go first), then Exposed Wounds, and that baby raptor would die. The next raptor would do Black Claw and Hunting Party and then it would die. By now Xu-Fu has got the double damage debuff from one hunting party application. The last raptor comes out and does Hunting party and Xu-Fu dies!
Any different team suggestions? Would love to hear from you so I can try them out!