I'm not Shulk expert, but I've been playing it since Smash Wii U came out and I think I'm pretty decent with him by now, so I'll try to explain to the best of my ability and according to my personal experience. As for the arts use, it really depends and you have to get used to think quickly of what's your best option to adapt for the situation.
Jump might not seem super useful at first, but you'll notice it's one of the deadliest Arts because it can help you gimp characters super easy. Also, when your opponents are reaching mid-high percents ranging from the 60s up, you can use jump to chase them in the air to keep the combos going. If you hit them really hard with a side smash, you can use Jump to go off-stage and finish them off. It also depends a lot on who you're facing. Jump is really useful for characters that like going high up to stay safeish, like Kirby and Jigglypuff, or what that dude who won the Invitational Tournament with ZSS did. I've noticed Diddy Kongs have a hard time with Jump, probably because the horizontal range of the jumps is so ridiculous it kinda makes it harder for them to set up their bananas accordingly. It's also conveniently the first Art in the reel, so if you get knocked off the stage in a diagonal angle, switch to Jump first, and second jump later. Most of the times you won't even need the Up B.
Speed is really hard to control if you're not used to speedy characters, so you have to practice a lot with it. I myself aren't very good at getting the timing right for the attacks and grabs yet, but the potential of the art is amazing. It can cut your distance short against trigger-happy characters like Link, Samus, Zelda (sometimes) and the Villager in a breeze, specially because your walking speed becomes almost, if not exactly as fast as your regular run distance. Walking makes it easier to perfect shield their projectiles while maintaining your movement, so it's a good thing to keep in mind. Speed is also very good for recovery because it increases your air speed. That is, how much horizontal movement you have in the air. It is mostly great to use against spammy characters and also characters that move very fast in order to keep up with them, like Sonic or Little Mac. The sheer speed also makes punishing from a distance really easy.
Shield is kinda bad. Sure, you get reduced knockback, which can DEFINITELY save your ass sometimes, but the big drawback that it reduces your speed makes it almost impossible for you to combo, and also, you become much easier to juggle as well, since you lose any kind of air speed Shulk may possess. Also, if you're unlucky enough to get launched far from the stage in this art, chances are that you'll die anyways over not reaching the stage horizontally and you're slow to activate Jump. I can't think of much uses for Shield. It also reduces the knockback you cause, which I guess could lead to better u-tilt chains but that's the extent of it. A shame, cause it's one of my favorite arts in Xenoblade :D
Buster is sooo good. A little bit too good. If you're unsure of what to use against someone, just stick to Buster. You probably know it increases the damage you deal. I'm not 100% sure of the exact amount but you should look for it somewhere in the Q&A thread and surely find the numbers. The thing about Buster is that you can't go wrong with it, but you have to make sure to string your combos well when you activate it to use it to the fullest. There's not much else to say about it. It can also help you sooo much if your opponent is in a fresh stock and you're some good chunk of % behind. You'll find yourself doing a lot of awesome comebacks with Shulk over the ridiculous amount of damage you can deal with Buster. Also, it is probably the most useful against the heavyweights like Bowser, Ganon and Dedede, mostly because it will take you less time to get them to dangerous damage levels.
Finally,
Smash is the finisher art. It boosts the knockback of your attacks so much that you'll find yourself killing people at ridiculously low %s sometimes. The other day I KO'd a Villager at like 15% by countering his Timber while using Smash. Once you have your opponent in the 100%s, just pull out Smash and chances are something you use will probably kill them. If it doesn't do it, the knockback will send them flying so far that you'll have time to switch to Jump for the easy chase or you can just stick to Smash to try and give it another try. Another good use for Smash before the opponents are in the 100's is that it can also be good to gimp recoveries early because of the added knockback. Characters like Link, Little Mac, Marth, Lucina, Ike, Mario and others that have a mostly vertical recovery will get really ****ed by a well placed aerial once they're off-stage. This is true even without Smash, but using the art will make early gimps almost foolproof.
Phew, sorry for the bible text, hopefully didn't bore you to death with this post
. I also didn't cover the disadvantages of each art, which is something you definitely wanna watch out for, but I'm pretty sure you can find that in the Q&A thread as well~
Good luck!