If you're looking here to figure out how to get the achievements, as of the most recent update (March, 2022) it seems the achievements are broken for most, if not everyone.

If you want to continue progressing through the game, consider keeping separate save slots before each ending trigger so you can come back and get them!

Apparently the issue with achievements has been fixed in another update. They claim watching the ending in the Extras menu may pop the achievement, otherwise you may need to complete it again.

If you haven't noticed or aren't familiar with Chrono Trigger, there are 13 achievements all corresponding to a different ending. I'll try to keep this as spoiler-light as possible but I'd advise beating the game naturally prior to trying to hunt for all the endings because (mild spoiler incoming)
some endings are only advisable in New Game + mode.

Since it was asked, and may be helpful for others, here's what I believe to be the most efficient order to get them all in two runs:

• Beyond Time (first run)
  
• Dream's Epilogue (first run is the most efficient if doable and sets you up for the second run)
  
• The Dream Project (if doable in your second run)
  
• The Successor of Guardia
  
• Good Night
  
• The Legendary Hero
  
• The Unknown Past
  
• People of the Times
  
• The Oath
  
• Dino Age
  
• What the Prophet Seeks
  
• Memory Lane
  
• Reunion

Thanks to Beldo Crowlen in the comments for these tips:

Note: As usual, completing the game before reading is advised.

To trigger any of the 'bucket' endings and skip the initial Lavos phase where he imitates the other bosses, you can either:

• Take the right teleporter at the Millenial Fair
  or
  
• Beat that first phase and save/return (there's a return teleporter there), which causes that phase to be skipped from then on

Chances are if you took my suggestion, you happened upon this ending naturally.

• Make your way through the Black Omen
  
• Use the bucket at the End of Time any time after Crono rejoins the party

• After beating the game, three dimensional vortices will unlock in 12,000 BC, 1,000 AD and 2,300 AD. Beat each of those dungeons, talk to Gaspar at the End of Time, then use the bucket and choose the new option, "Time's Eclipse"

• Defeat Lavos at the Ocean Palace in 12,000BC
  
• At the Millenial Fair when Lucca is showing off her machine instead of engaging as normal, use the right teleporter. New Game + only

New Game + only

• Similar to the previous ending, use the right teleporter right after finishing up in 600 AD.

New Game + only unless you're insane

• Use the bucket at the End of Time right after arriving for the first time.

New Game + only unless you're insane

• Immediately after arriving in 600 AD the second time, go back to the End of Time and use the bucket.

New Game + only unless you're insane.

• Use the bucket at the End of Time right after finishing up in 600 AD the second time.

New Game + only unless you're insane.

• Use the bucket at the End of Time right after getting the gate key back.

New Game + only unless you're crazy.

• Use the bucket at the End of Time immediately after giving Frog the repaired Masamune.

Most likely New Game + only.

• Use the bucket at the End of Time after fighting Magus but before finishing 65,000,000 BC.

• Use the bucket at the End of Time as soon as possible after finishing 65,000,000 BC.

• Use the bucket at the End of Time after Schala opens the sealed door but prior to restoring Marle's pendant.

Easier to accomplish in New Game +

• Defeat Lavos after losing in the Ocean Palace but before re-acquiring Crono

This ending is easy to achieve (though there's no achievement for it). Just lose any battle that can be won.

Not much to see here but it is different than losing anywhere else.

• Lose to Magus during the first battle with him

To summarize this ending, You die, Magus turns around, you hear Lavos, game over.