I'm going to describe the battle system in terms of the Classic Controller. The Nunchuk is also a method of playing but tbh I've never used it so idk how it works. If you've never used the CC, it's basically a GameCube controller with ZL and ZR buttons, and the l-stick and d-pad are switched.
BASIC BATTLE STUFF
During battle, you move your character with the l-stick and control the camera with the r-stick. Getting close to a targeted enemy will make you autoattack it (there is one character in this game that uses a gun, and therefore has ranged autoattacks but that's not a big deal here). Targeting enemies is simple: if you see a box above their head with information like the name of the enemy in it, that's the enemy you've targeted. You can change targets by using the L and R buttons - and on the CC, these are like the Z button of a GameCube controller, light taps are sufficient. The targeting is kind of iffy in this game (Pressing L and then R in a large group of enemies might not refocus on the one you originally were targeting) but that's not a huge deal. You fight alongside 2 CPUs that are controlled by the AI with varying degrees of effectiveness.
ARTS
The dpad is used to select your "Arts", this is a palette of circles at the bottom of the screen that are basically the abilities you can use. Each Art does something different, whether it's inflicting damage, a status condition, or buffing you or your teammates. Arts also have cooldowns, similar to Shulk's Monado Arts in Smash: after you use one, you can't use it again until a certain period of time has elapsed. The exception to this is the middle, larger circle - your Talent Art. This is generally raised by autoattacking, although there are some exceptions. If an Art is grayed out, it is not fully charged and you can't use it. Arts that have lit up can be used by selecting them with your d-pad and pressing A.
Generally Arts are split into several categories depending on their color:
Red indicates that an Art is a physical attack. These tend to inflict a debuff on the enemy such as slowing them down or lowering their defense; alternatively the Art might deal extra damage based on certain conditions such as you positiong relative to the enemy.
Purple indicates that an Art is an ether attack - Ether is the "magic" equivalent in this game. Ether attacks generally inflict some sort of status condition.
Blue indicates that an Art doesn't do damage and instead offer support to you or a teammate - this could be through healing or buffs (such as raising Physical Damage Dealt).
Pink,
Green, and
Yellow indicate that an Art inflicts a special type of status: Break, Topple, or Daze respectively.
Orange indicates an Aura - these are Arts that apply a certain effect for a certain period of time - much like Monado Arts or Lightweight in Smash Bros. These are generally very strategic, such as raising attack while lowering defense, or increasing dodge chance while lowering accuracy or something. You can only have one active at a time.
OTHER RANDOM STUFF
There's actually a lot more but I doubt you'll want to read through it so I'll give a brief summary:
Aggro: An enemy will lock onto the party member that has drawn the most aggro and will focus all of its atacks on that party member. Aggro is raised by attacking enemies, and can be lowered by distancing yourself or using specific Arts.
Break, Topple, and Daze: Special status conditions. Break does nothing but will allow Topple to be inflicted, which in turn will allow Daze to be inflicted. If an enemy is Toppled or Dazed, it is not able to attack and aggro will not be drawn so you can just whale on your enemy without any repercussions
Party Gauge: Fills up when you inflict a critical hit or activate an Art's special effect. You can use a third of it to revive a fallen party member, or use all of it when it is completely filled to unleash a Chain Attack
Chain Attack: You cycle through each of your party members and choose an Art from each to use, without cooldown penalty. This is a great time to inflict Topple/Daze; however if you use Arts of the same color consecutively the power of each successive Art will increase. After each party member has gone, you will have a chance to go back to the beginning and use another Art through a reaction cue - get it right and you can use another Art; this is commonly known as a Chain Link. This can continue ad infinitum afaik.
Burst Affinity: A reaction cue that will allow the bond between two characters to grow - this raises the chance of a Chain Link happening and has many other benefits ouside of battle.
Tension: Tension is raised by successfully landing attacks and blocking/dodging enemy attacks, and lowered by the reverse. Tension affects a character's strength and accuracy. There are 5 stages of tension - if a party member falls to the 1st or 2nd stages (assuming 5th is the best, and 3rd is the stage you start on) they can be encouraged by going up to them and pressing B.
Visions: Occasionally the screen will go black-and-white and an enemy will be seen doing a strong attack on one of your party members. A timer then appears at the top of the screen during which you can use a third of your Party Gauge to "warn" a teammate: you can then use any of their Arts without cooldown penalty. This can be used to heal, redirect aggro, raise defenses, or just flat out killing the enemy in order to change the future.
That's it for your crash course on Xenoblade Battles :D
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don't even need to get the Chain Link
. Sometimes in battle I'll zoom in if I'm really feeling 
You may want to look somewhere less biased.
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