I finally started this game after many months of leaving it to gather dust.

It is really amazing, but I don't know what starts chains, or how to execute them.

Chain attacks can be really great. Not absolutely necessary, but fun and helpful. It has been a while since I played it originally, but as I recall you'll gradually accumulate a "Chain Gauge." Once that fills up, the battle icon used to open up your Monado options.... Or just the center I guess, if you're controlling a different character, can be switched to an icon for chain attacks. Activate it when you get the chance. Okay, now it gets more fun and complicated. To increase the chain level and damage multiplier, you'll need to use consecutive attacks with the same color. Green, red, pink, blue, whatever, you have to chain them together across characters, going round and round in a cycle starting with the character that you control. You also have to hit a timed button to prolong the chain as you go. Here's the key: White color attacks can be used to transition between colors. For example, if you run off a string of reds, you can use a white attack and then transition to green or pink with the next attack and go from there. Once you use a skill, it is grayed out for cool down. Set up your character abilities accordingly to allow for big chains... It is the best way to drop big damage on bosses, plus there are some achievements associated with it if you're into those.

yup, what @Tackchevy said.

fill up the team gauge by attacking the enemy, the gauge is separated into three bars. when all three are full, you'll see the blue strings connecting you to your party members (it also should show a message or have someone say "it's time for a chain attack" or something). then you go to your talent in the middle (the big circle) and push down once or twice (until you see the blue colored circle with the white triangle on it, I think before that there's the "flee" option which is green). you select that which initiates the chain attack. aside from the colors of the types of attack you choose, you should also mind the positioning of your character relative to the enemy (in case you wanna use an attack that has increased damage when hitting the back or side of the enemy)