-Don't let squishy members like Shulk and Sharla accumulate too much aggro. Characters like Reyn and Dunban instead need to draw it (Reyn defends, Dunban dodges). If you see a ring around Shulk, ease off on the Arts (or use Shadow Eye) to reduce aggro.
-Don't spam Arts unnecessarily
-Healing Arts accumulate a ton of aggro early on.
-Don't aggro too many enemies at once, you will likely be overwhelmed regardless of tactics early on, lol
-Take advantage of the Break/Topple/Daze system. It works in that order. Easiest way to identify those Arts are Pink/Green/Yellow (not orangey, but pure yellow) respectively. Those colour icons always mean to inflict those attempts of abilities.
-When you see the future, respond appropriately. Ironically the first thing the game gives you is Shield, to block Talent Arts (white), yet early on enemies more or less seem to use Physical Arts (red). That's usually better responded with Monado Speed (if you've gotten it). Make sure to take advantage of the Warning system too. IE: If you need to change the future, have say Shulk use a Break skill, then warn Reyn to use Wild Down to Topple the enemy (that ensures he does it, as opposed to the AI possibly choosing something else before Wild Down)
-Understand each character's unique play styles. IE: Shulk is all about positioning (Front/Side/Back) to utilize hits Arts bonus traits.
Edit:
Oh the Chain Link system works like this
When you use an Art, the gauge fills. When you achieve Burst Affinity (basically when a character calls out and the circle goes around the B button for you to press), that increases the gauge (and also builds affinity, so yay).
If you fully fill all three bars, you can use a Chain attack (and depending on how heated up characters are (look at their portraits, their faces can be normal, depressed, heated [actiony], or heated + on fire), that can affect the length of chain attacks.
Of course the Chain Link gauge is also required for reviving KO'd party members (and if you ever have less than one full bar and you leader is KO'd, the fight is loss), as well as to Warn other party members to change the future.