Getting the full experience from Chrono Trigger specifically, unlike most other similar games, is getting all of the endings. The New Game+ mode will help there.

Chrono Trigger has 18 endings, if I recall correctly, including various permutations. Plus one additional one in the rereleases from the DS version forward. Some of them are only very subtly different from each other depending on which combination of character side quests you fully completed, and they all vary depending at which point your manage to defeat the final boss in the main story sequence. Several of them are significantly different interpretations of the future (or the past) post the defeat of the final boss with various for-want-of-a-nail factors causing huge changes to the outcome.

You have quite a few opportunities to fight the final boss up to and including immediately after discovering the first time gate all the way back at the beginning of the game. (Do that in New Game+ with an overpowered Crono, though, unless you want to get steamrolled instantly...)