From the first room, enter the door to the right, then turn left and head up the staircase. From there head left, and then turn and head left again. You'll be at a platform where you can see the first room. To your left, there is a small bounce pad which you can use to launch yourself atop of a pillar. Jump over the pillars to reach a Super Health.
As you enter the yard with the elephant statue, head left and you will see a small health pill next to a wall. Pick it up, and a few Bio-mechanoids will be spawned, and the wall will be blown up, allowing you passage to a secret yard. In SSHD, the wall must be manually blown up using a rocket.
From #2, head to the end of the secret yard and look for a detonator switch: activate it to blow up one of the towers and reveal some secret cannonballs. In SSHD, a Serious Speed will spawn.
On the left-side wall of the building on the right of the elephant yard, there is a darker section on the wall which you can blast. Do so and collect an armor piece. (once you do, it will spawn an armor shard which in turn will spawn a few Major Bio-mechanoids.
After leaving the elephant yard, immediately take out every enemy in the small yard. Afterwards, 6 Kleer skeletons will be catapulted to you from the wall on the right. Destroy them while they are still in mid-air to get a congratulatory message. The RAPTOR Sniper Rifle is recommended for this situation.
In the following yard from #5, head to the bottom right corner of it, next to the gate which you exited from, and pick up some ammo. This will spawn a "mutated plant" (bio-mechanoid).
Opposite of #6, there is a darker section on the wall which you can blast to reveal a +100 armor.
Diagonally opposite of #6, or just head north from #7 and then take a turn left, there is a heart in a darkened opening to the wall.
At the yard where you see the golden elephant for the first time, make a whole circle around the yard and find a switch attached to the wall. Activate it to enable a secret yard.
At the yard with the minigun, there are four doors in total: the one you entered from, the one that leads to the golden elephant, the one that leads to the next areas of the level and a locked one. After using the switch from secret #9, proceed to the locked door to find the secret yard. Deal with the initial Kleer wave there and walk towards the end of the yard to find a secret supplies depot.
(HD-only) In the building accessed after acquiring the golden elephant, you'll notice there's a room with a spinning platform in the middle. Look above the platform, high in the left or right side of the wall (depending on where you are standing), and you'll notice a Super Armor. Then, look below the platform to spot a bouncer, which you can use to be launched to the armor. This requires timing and patience to do so though.
At the final yard, before heading up the staircase, to the left of it is a switch. Use it, and then head to the right, past the exit of the level, and enter a locked building for some supplies.
Still at the final yard, use jumping to get behind the horse, until you face its bottom, and then press the Use button to activate the secret "horse power" (spawns a Super Health).