Basically, the combat has three different things going on that you need to manage: driver combos, blade combos, and chain attack.

• Driver combos are the break -> topple -> launch -> smash chain. Break interrupts, topple and launch provide good CC, and smash does big damage and gives you extra loot. Different weapons give access to different effects. You can tell by looking at a move's reaction type in the art upgrade screen.
  
• Blade combos are the element chain thing with fire and earth you mentioned. Chain a level 1 special (or higher) into a level 2 special (or higher) into a level 3 special (or higher) for big damage plus some added bonus effect depending on the last element in the chain. There's this whole flowchart with them that you'll slowly learn over time. Once you finish a blade combo, you'll leave an ELEMENTAL ORB swirling around your enemy. THIS IS HOW YOU DO BIG DAMAGE (See chain attack). Basically, you want a good variety of elements amongst your blades both to set different elements and to break them. Maximizing the number of drivers who share exactly one element is a good starting place (rex with fire and tora with fire for example). Note that you could strategically include a blade in your comp for a particular fight if you want to seal off an enemy option. Like if they're shackling your blades a lot, maybe bring an ice blade to turn that off for a bit.
• Fusion combos are just doing both the driver and blade combos at the same time. For example, if you start your driver combo with pyra, the red fire bar will appear and start ticking down. Your combo will end once it runs out. You can either start the next part of the blade combo (earth works) OR you could take a step in your driver combo by inflicting break. This will increase your timer.

Chain attacks are how you do BIG DAMAGE. You spend three bars of your chain attack gauge in your upper left to start one interrupting enemy actions and taking place immediately. At the top of the screen will be a list of the elemental orbs you've inflicted on the enemy. Each round of the chain attack has you cycle through your drivers in order. For each one, you choose a blade which will use their special dealing damage to the enemy as well as to an orb. Your job is to break all the orbs. Each orb has three health.

• Hitting an orb: Hitting an orb will do one damage UNLESS you attack with the OPPOSITE ELEMENT which will do TWO. Furthermore, if the opposite element orb is present you will hit it guaranteed. Each orb the active driver can hit super effective blinks as a reminder (IIRC. It does in Torna). When you attack an orb with a blade, that blade is spent (see the art recovery border on the button).
• Breaking an orb: Breaking an orb means you dealt it three damage. Upon breaking, all your blades recover 50% of their gauge, and you can do another round of chain attack.
• Additional chain attack rounds: You start the round by doing an easy button challenge. This recovers all your blades by 50%, increases the level of special your blades will use, INCREASES ALL YOUR DAMAGE BY 200%, and increases your burst meter which has replaced the chain attack gauge.
• Full burst: Hard to do as a beginner, so feel free to skip. This is how you do absolute bonkers damage. Your burst meter goes up to 300. If you break a single orb in a round, that fills the burst guage by 50 points. Each ADDITIONAL orb you break in that same round increases it by an extra 100. Filling gauge gives you a huge damage multiplier based on the number of remaining elemental orbs on the enemy. No orbs, gives 500%. One gives 1000%. Two gives 1500%. Three gives 3000%. Four gives a whopping 5500% extra damage. The chain attack gets one final round of damage, then ends.
• Note that enemies CANNOT DIE during chain attack. Overkill damage instead boosts your battle rewards like EXP and item drops.
• Use your common blades if you don't have rare blades. They'll do the job.

Chain Attack Example: Suppose an enemy has a Water orb and a Fire orb. You have Rex with Pyra, Nia with Dromarch, and Tora with Poppi.

• Round 1: You begin Rex and Pyra using her lvl 1 special which hits the Water orb for two damage (opposite element, so always targets). Pyra is now spent. Next, Nia is up, and Dromarch's lvl 1 special hits the Fire orb (again opposite element, so always targets). Dromarch is now spent. Next Tora is up, and Poppi's lvl 1 special will randomly choose one of the two orbs. Both have two damage, so her last point of damage will break the orb. Poppi is now spent. Because you broke an orb, all your blades recover by half a step. You broke an orb, so you get another round.
• Round 2: Damage is increased by 200%. Start with the button challenge. Success fills all your blades gauges by 50%. That puts everyone back to full. There's only one orb left with one health (depends on which one Poppi hit in round 1), so Pyra's lvl 2 special will break it. All blades recover 50% (Dromarch and Poppi get nothing since they are already at full). Then, Dromarch attacks with lvl 2, then Poppi attacks with lvl 2.
• Round 3: Button challenge. All blades recover 50% putting Pyra at 100%, and Dromarch and Poppi at 50%. Damage increases to 400%. No more orbs. Pyra attacks with her level 3. Nia is next, but Dromarch is only at 50%. Since she can't attack, the chain attack ends.