With everyone and their mother on here maining Shulk, I figured that I'd try giving Shulk a a guide and maybe learn what exactly made everyone on here go for him. Along the way, I also got some tips from KnowledgeablePig, who's got a pretty good Shulk, on how to play him. The result was this incredible wealth of "knowledge" and sarcasm you see before you.
So without further ado, here's a probably not so great guide to being the Monado Boy your mom always wanted you to be.
The biggest difference between Shulk and the rest of the cast is the Monado Arts. By pressing B, Shulk can drastically alter the properties of his attacks and his person. Each different art buffs and nerfs some of Shulk's abilities. Succeeding with Shulk depends significantly on using said arts effectively, and by extension, the biggest key to beating a Shulk is knowing how to approach each form Shulk differently.
So before we move on, here's a rather useful chart of what buffs and nerfs each Monado art gives Shulk.
Depending on the circumstances of the fight, different Monado arts will be useful at different points. So to help with that, I put together a chart of which (I think, anyway) Monado Arts excel where.
Jump: Jump's generally not useful unless you're off stage and needing a boost to recover. You can quickly hit "B," and then take advantage of the extra height on your jumps and Up B to recover from situations you normally might not. If you've got an opponent off stage, you can also activate jump and go after them much deeper than you normally would since your recovery is boosted.
Speed: Some Shulks rely almost entirely on Speed. I, personally, think this is a mistake. Speed is useful when your opponent is at lower %s as you can combo much easier. After your opponent reaches 50% or so, it's not worth it, as opponents can get out of strings much easier. It also has some use on getting in on opponents who do a good job of keeping Shulk out (such as Robin or Ness).
Shield: Shield basically turns Shulk into a brick. Unless you're playing someone like Ganondorf or at ridiculously high %s, the only way you're dying is if you can't recover. Shield, as such, is a good tool for prolonging a stock. This can be used to horribly frustrate your opponent since they frequently lose their ability to kill you.
Buster: Buster turns Shulk into one of the most damaging characters in the game. Simple attacks like tilts or throws do around 20% damage in this mode. The big drawback, though, is you basically can't kill anyone. Buster, then is useful for wracking up extra damage on opponent's not in kill range.
Smash: Smash is the mother of all gambles if both you and your opponent are in killing range. Mind games become real here, as your arsenal of killing attacks expands dramatically (forward tilt and down throw become real threats to kill). I'm not a huge fan of this one unless I'm not in kill %s and my opponent is, mostly because I'm terrible at Shulk mind games. Pig's fantastic at them though, so there's certainly room to use them if you don't suck like me.
Keep in mind, depending on your play style and your opponent, you may want to use these differently. Some, though, are pretty universal (never use Shield when you're at low %s, never use Smash when your opponent isn't in killing %s).
With all of that said, I'd argue that the most important piece of playing Shulk well is using his aerials correctly. Shulk has some of the best aerials in the game; they're long duration, cover quite a bit of space, and are pretty safe if you use them well. In fact, I'd argue they're so important that I'm dedicating an entire section to them.
Neutral Air (simply pressing "A" while in the air), causes Shulk to swing his sword upwards in the direction he's looking. This is a really good approach tool (it's nigh on impossible to punish if spaced correctly), excellent combo opener, and will generally beat out just about any aerial your opponent tries to counter it with. It also eats most single hit projectiles, so you can use it to safely get in on characters like Pikachu and Samus who are zoning.
Forward Air causes Shulk to swing his sword downward in the direction he's facing. This is an even better combo tool than Nair, but the drawback is that it's never safe if your opponent shields it. Like Nair, it will also eat projectiles and beat out just about any other aerial.
Back Air has Shulk sticking the Monado out behind him, covering an almost ridiculous range. This is a fantastic kill move and an excellent change of pace that can catch people off guard. Due to its insane range, it's also really hard to punish if you space it right.
Up Air and Down Air aren't quite as good, both involve Shulk sticking the Monado above or below him with a ton of recovery time. Both are good kill moves, for what it's worth.
If you get these two things (Monado Arts and Aerials) down, Shulk's pretty easy to pick up and play. The final piece to getting him down is getting used to his counter, which is pretty easy. The counter has an incredible window of opportunity, and its speed can be altered as well by pressing the direction you want to counter when the animation triggers.
Couple final things of note:
Shulk's Side Special (Backslash) works well as a change of pace. It gets very predictable and easy to punish if you use it frequently, though.
Air Slash has two parts to it; an initial upwards slash, and then a horizontal slash. You will only get the horizontal slash if you press B twice, however. The second slash can kill at higher %s, but it's extremely risky.
Shulk's pivots can be devastating. If you run past an opponent, and pivot back with a side smash or f-tilt in smash, your opponent can die really quickly. Shulk's smash attacks are extremely powerful in general and have some pretty crazy range...but they also have a ton of recovery as well, so don't go too crazy.
Shulk's Down Smash works pretty well as an edge guarding tool. It lasts longer than the invincibility frames from grabbing the edge, so you can use this to punish an opponent who doesn't get up immediately.
That's all I've got. Hopefully this helps you show the Monado's power to everyone around.