Summary
To perform a Forward Dash, you have two options:
- Tap Towards on the controller twice in a row rapidly.
- Pressing any two of the three main Attack Buttons while the controller is Neutral or while holding the Towards direction.
To perform a Back Dash, you have the same two options:
- Tap Back on the controller twice in a row rapidly.
- Pressing any two of the three main Attack Buttons while holding the Back direction.
Please note that Holding Down/Toward or Down/Back will not allow you to dash via the buttons.
A Dash of Mobility
Dashing is a very important part of Marvel Vs. Capcom 3. Mobility is key in every Versus Game, and, while on the ground, Dashing will be your main source of mobility while on the ground. Dashes move your character very quickly in the direction you are Dashing. This allows you to traverse the screen on the ground much more quickly so you can either close the gap on your opponent or quickly get away from them.
Some characters have slightly modified Dashes on the ground. Morrigan's Dash takes her into the air at about a 30% angle, so she gains access to her Air Moves instead of her ground moves. This happens on both the Forward and Back Dash. Hsien-Ko's Back Dash behaves as you would expect, but her Forward Dash behaves exactly like her Dash from the Vampire series: she disappears briefly and teleports forward a fixed, short distance where she then reappears. As long as you keep holding the controller forward, she'll keep teleporting forward in a slow pace, where she reappears and then disappears again to move farther forward. During the portion in which she disappears, she actually is invincible and can pass through attacks. And finally, Arthur simply does not have the ability to Dash at all!
Canceling Your Dash
What makes Dashes powerful is that they can be canceled before they complete. There are three ways to cancel a Dash: with attacks, with a Crouch, and with a Jump. When you cancel a Dash with a Normal Move, you gain the extra benefit of maintaining the momentum of the Dash with your attack. That way, if you get in on your opponent and start performing a series of Normal Attacks on them from a Dash, you will be able to land more hits on the enemy before being pushed out of range thanks to that momentum. When you cancel a Dash into a Special Move or a Hyper Combo, however, the move does not gain any extra momentum and always behaves the same as if you performed it while standing still. And when you cancel a Dash with a Crouch, you'll slide forward a tiny bit before coming to a stop. Obviously, Morrigan is the one exception that cannot cancel her Dash into a Crouch.
With attacking and crouching, however, different characters have different windows before they are allowed to actually cancel the Dash. For example, X-23's Dash can be canceled into a Normal Attack or Crouch almost instantaneously. Chris, however, cannot cancel his Dash until after a short period of time, when his front foot becomes planted again. Ryu has an even larger window than Chris before he can cancel his Dash into an Attack or a Crouch. Then there are characters like Dormammu, who cannot cancel their Dash into a crouch or an attack at all.
Strategy Corner
Taking advantage of a Dash's ability to be canceled into a Crouch allows players to perform what has come to be known as "Wave-Dashing." This technique existed in MvC2 as well and was widely utilized as a means to travel the fastest you possibly could travel across the ground. The way it works is this:
Whenever you perform a Dash, usually it's the start of the Dash that has the most speed and momentum. When you 
it into a Crouch, you do slide forward a bit as mentioned earlier. However, you are still grounded so now, technically, you can do another Dash! Because you can dash forward just by pressing two of the three main Attack Buttons, the easiest way to dash again is to simply let go of the controller and press the two Attack Buttons.
So now, if you learn the proper rhythm for your character, as each character's rhythm is slightly different, you can alternate between pressing Down on the controller and pressing two of the three Attack Buttons and what you'll end up doing is repeatedly starting your Dash and immediately canceling it with a Crouch and the immediately performing yet another Dash. So you end up stringing together the fastest part of your Dash repeatedly in a very short amount of time. This allows you to travel across the screen very quickly, allowing you to take better advantage of Knock-Downs and moments when you have the chance to try to close in on your opponent. It's a very powerful technique when used properly in the right places, so don't overlook this technique.
However, when it comes to the third type of canceling, every character can cancel their Dash into a Jump at any point of their Dash with no restrictions (except for, once again, Morrigan, who goes into the air). Also, when you cancel a Forward Dash with a Forward Jump, you also gain a momentum boost on your Jump so that your regular Jump travels a father distance as well. However, Bask Dashing and performing a Back Jump does not make you jump backwards farther.
Downsides to Dashing
Dashing comes at a small price. Whenever you perform a Dash, whether by tapping the controller or hitting two buttons, your character will be committed to that Dash for a short period of time. That means if the opponent is, for example, Ryu and activates a Shinku Hadoken Hyper Combo from across the screen right when you perform a Dash, the Hyper Combo will connect on you if you try and block.
This is different than from MvC2. In the previous game, you could cancel a Dash into a Block only if you performed the Dash via the method of pressing the two buttons. So even if Cable performed an Air Hyper Viper Beam, which hits almost instantly, during your Dash, you could stop and Block just by holding Back on the controller during the Hyper Combo freeze.
However, there is a way to duplicate that in Marvel Vs. Capcom 3! Even though you are committed to your Dash, you can use a technique to get around this restriction. In that scenario I listed earlier, where Ryu activates that Shinku Hadoken right when you started your Dash, it's true if you try and hold Back you will get hit. However, if you hold Back and Up at the same time, you'll cancel your Dash into a Jump and then, during the Pre-Jump frames right before your character leaves the ground, the Jump gets canceled into blocking on the ground! That will actually allow you to Block the Shinku Hadoken from Ryu.
What that means is that you can pretty much block anything during a Dash. So if you perform a Dash right when Spencer performs the Bionic Lancer, which hits almost instantly, you can hold up and back to block it successfully after the Hyper Combo activation flash. You definitely have to remember this tactic, or you'll find yourself dashing into and eating Hyper Combos left and right.
Morrigan
A side note about Morrigan's Dash: when she takes to the air, there is a tiny window at the start that you cannot cancel into anything, but after that it can be canceled into whatever you want in the air. Most notably, you can cancel into an Air Block very easily, so if anyone ever activates anything while you are Dashing from the ground with Morrigan, you can always block in time.