The first section of the Menagerie has you lighting lamps ablaze. You'll need to take down special Lamplighter Knights and they will drop orbs. Take those orbs to the lamps that are circled around the middle treasure area and they will light up and add to the percentage you need to complete. Do enough of these and it will unlock the next area.
There's a lore item in the middle of this room, it's a big Vase with a bunch of treasure surrounding it. This one is pretty crazy, a bunch of different Fallen enemies will spawn that you'll have to lay waste to. Try to stick with your team as much as possible, and pick up any downed teammates because people will likely be getting killed left and right due to the amount of enemies there are. The object of this one is to slay as many Tamed Knights as you can. The more you destroy, the more the percentage you'll get towards completing The Menagerie. You have a certain amount of time to complete this, so try to kill as many as possible before it ends.
There are other knights called Cherished Darkblades you can kill, and when you do they will drop an ancient relic sword. You can use this to deal some pretty heavy damage to those Tamed Knights and other enemies. The Crystals is kind of a small room with a lot of vertical space to it. You'll be killing various Vex enemies, and eventually Void Cannons will spawn. You need to grab one of these and find one of the purple crystals that are randomly spawned on the walls of this room. Use the Void Cannon to destroy the crystals, and you will gain points towards completing this room. The Void Cannon can also be used to meltdown enemies as well, but they have a very limited supply of ammunition. The Arkborn requires you to gather up Arkborn Souls that spawn around the room in the form of glowing yellow light. Run through these and you will gather up the souls, bring them over to the yellow bonfire on either side of the room. You only have a limited amount of time to turn these in, so do it quickly (you can carry multiple at a time). At some point during this round, a big Treasure Ogre will spawn and the bonfires and souls will be removed from the room. You'll need to destroy the Ogre before going back to collecting and turning in souls. Don't mistake the souls for the similar looking yellow traps that get scattered around the room. You are likely familiar with the look of these, but the traps kind of look like small bonfires, and the souls are just beams of light. In The Hunted you will need to locate a glowing circle in one of the four corners of the map. Your team will need to sit in this area and defend it by slaying tons of Fallen enemies. This includes a pretty tough Treasure Knight that will spawn and attempt to kill you. It has a ton of health and will pretty much one shot you if you get too close. Once you've fully captured one of the circles, another one will spawn and you will need to move onto that one. There will be a field in the room that you and your team will need to stand in. Kill the Wizards that spawn around the area, they will drop orbs that you will have to go retrieve and bring back to the safe area to place into the receptacle that's in the field. You start off with killing some monsters and a boss. Then you'll run the actual gauntlet, which has you running a lap around an area that has a bunch of obstacles.
There's pits to fall into and there's some jets of flame that will attempt to kill you along the way. Avoid all of these and make it across the finish line. You then do another round of killing monsters, and another round in the gauntlet. Once you've gathered up enough percentage towards The Menagerie percentage bar, you will then be tasked with fighting a boss battle.
This is a huge Vex boss with an impenetrable shield around it. To be able to damage that shield, you'll need to kill Hasapiko's Living Firewalls that spawn. Once you kill one of these, it will leave behind a white glowing area that you can stand in and that allows you to deal damage to the boss. If you deal enough damage to the boss's shield, they will become vulnerable to damage.