After defeating all of the suits of armor in the arena, the first thing to do is climb over the wall on the right side.
Follow the edge of the arena around, and you'll find a Trico Treat. Toss it over to Trico.
Trico will take an interest in a wheel mounted in the wall above the doors. Your job is to encourage him. Ask him to jump to get him to stand against the wall (you don't have to be on his back to get him to do this). This will give him the leverage he needs to spin the wheel and open the doors.
In the room on the left, you'll find a cart. Pull the cart, out and position it in the sun below the grate high above you. Stand on the low end of the cart and wait. Trico will come over and paw at the other end. Eventually, he'll stand on it, sending you flying.
Climb up through the grate and start throwing switches to open doors. Head out onto the balcony over where you arrived at the arena and call to Trico (we're not giving a direction to turn here because your orientation can and will vary — you'll know it when you see it). He'll hop up and join you.
Go back inside and lead Trico out to the right.
Lead him along the roof to the left to the far end. Climb on his back and ask him to jump. He'll carry you up above the waterfall.
Go inside the room you find there. At the far end, on the right, you'll find a Trico Treat. Look above you for a gap in the railing. Direct Trico to jump through that gap to find the next room.
Trico again decides he doesn't like water, so you're going to have to start this one on your own.
Hop down into the water and swim toward the gate on your left. Dive under the water and look for a tunnel directly under the gate. Swim through, climb the ladder and throw the switch. Swim back out to the main room and call to Trico.
When he jumps down into the water, climb on his back. Direct him toward the gate and he'll eventually dive down — deeper than you can swim on your own — and find another tunnel that will carry you to the next room.
This room is mostly identical to the last one. Swim toward the gate, then dive into the tunnel below. The problem is that the ladder is broken. Tread water near the broken ladder and turn your attention to Trico.
Wait for him to climb onto the rocks on the far side of the room, then call to him. His entry into the water isn't exactly Olympics-worthy. The wave he creates will carry you up to the switch platform.
Once again, point Trico toward the gate and he'll dive down. It doesn't go well for you.
When you come to (that's number five, we think), make your way up the tunnel, where you'll find a room with two mechanisms in the floor. This starts a fairly long sequence, so we'll give it its own section next.