After collecting every upgrade from every stage, but before going into the Sigma stages, go back to Armored Armadillo's stage. If you don't have 9 lives, stay at the beginning of the stage and kill the bat enemy that looks like the ones from Mega Man 2 (and not like the rest in the area). This particular bat is very likely to drop a 1-up when you kill it, so you can keep killing it, then walk off and back on screen until you're at the maximum of 9 lives.
After you have max lives, play through the stage as normal until you reach the 3rd and final wheel platform that X rides on. I recommend equipping a charged shield for this to negate damage from enemies. Ride it to the end until you're soaring through the air. At the apex of the platform's arc, dash jump to the right. You'll land on or just below (wall kick up) a ledge with a big power pellet on it. Grab this and jump to your death to the right. Don't enter the boss door!
Repeat this process from the checkpoint until there's suddenly an upgrade capsule on the ledge. When you activate it, Dr. Light will be wearing a headband, and he'll teach you the Hadoken, the signature projectile attack of Street Fighter's Ryu and Ken! To use the move, have no special weapons selected and enter down, down-forward and forward on the d-pad, or in other words, slide your thumb from down to left or right (depending on the direction you're facing) and press the fire button.
The Hadoken is extremely powerful, and it can take down almost any enemy in the game in one hit, bosses included. There is a catch, though - you have to be at full health to use it. Besides that, the projectile is a bit slow, so it can be tough to hit enemies that move around a lot. I personally have a very hard time using it on the spider boss in one of the Sigma stages, which is a shame, because of all bosses I'd want to take down quickly, that metal waste of space is the worst. It's much easier to kill Sigma and his dog, but the final form of Sigma is sadly immune to the attack. You'll just have to stick with Rolling Shield for that one!
This move is a lot of fun, but there is a similar secret in the sequel, and I just got such a huge alliteration bonus... score!
In the third X Hunter stage, you'll reach a point where you see a ladder down, but also a ladder up, which you can't quite reach. Well if you've once again gathered every hidden upgrade throughout the Maverick stages, you can proceed. Lure a bat in from the left, freeze it with your ice weapon and leap from the ice cube to the ladder above.
Now you're in for some tricky platforming. Try your best to not take damage here, for two reasons. Reason one is that you can't trigger the secret if you don't have full health, and reason two is that most obstacles in this area will kill you instantly anyways. It's a spiky death trap! At the very least they provide you with a 1-up in the open so you can try as many times as you need.
The first step is to charge up a fire special. Once you're charged fully, you can jump down. Before you hit the spike pit, let loose your weapon and shoot across to solid ground. Take out the two wheel enemies before they run over you, then hop up up the steps. Here comes the hardest part. Charge up your fire weapon again. Then, you'll need to jump up and airdash to the right, and before you hit the spike wall, let yourself drop straight down and then release your weapon so you fly to the left. If done correctly, you'll land safely and win another 1-up.
From here, there are no more obstacles. Simply slide down the left wall and eventually you'll pass through into a room containing a secret capsule. Remember, if it isn't there, make sure you're at full health and have collected every other upgrade beforehand. If done correctly, Dr. Light will make a brief Wayne's World reference then teach you another Street Fighter signature move: the dragon punch, Shoryuken!
This move is similar in destructive force to the Hadoken, but instead of a projectile, it's a rising uppercut which sends X far higher than his normal jump can take him. This makes it even harder to hit enemies, but conveniently, in the stage where you find the Shoryuken, the stage boss starts the battle above you, so you can use it to dispatch him instantly. To use the Shoryuken, the input is forward, down, then down-forward, then shoot. Basically, forward, then the same input as Hadoken, but stop short of hitting forward again. Like the Hadoken, this is the same input as in the Street Fighter games.
It's quite sad that in Mega Man X3, you don't learn another Street Fighter move. They could have given you the hurricane kick, or maybe a move from another character, like M. Bison's Psycho Crusher or virtually anything Zangief does. Could you imagine X with Dhalsim's stretchy arms, or warping around the stage like Akuma?
If you took out Vile in his secret stage with his weakness, the miniboss in the 2nd Doppler stage will be different. Swap to Zero before entering the door and defeat it. Upon its defeat, it will come crashing down on top of Zero, who just kind of stands there and takes it instead of using one of his many abilities or attacks to even attempt to dodge the big robot falling slowly towards him.
Anyways, X will rush in and Zero will explain that he took serious damage and needs to return to base. Before he leaves though, he lends X his Beam Saber. Now, X can attack the same way Zero does, with a charge shot resulting in a 3 hit combo ending in a sword slash. The sword is super strong, but still not as strong as the previous two specials. Apparently if you don't have the arm upgrade, you won't even do the combo, and you'll just rapid fire sword slashes. Talk about destructive.