Chrono Cross for me is a brilliant game, and I love it just as much as Trigger for its own reasons. It does have its flaws, but I really would not change too much overall.
One change I would make is providing a clearer picture of what happened to Crono and Marle after the Fall of Guardia. It is still left ambiguous and while they appear to be dead in Home World, it is suggested they are alive in Another World. However there are no hints to what happened to them after the fall outside of ambiguous factors here and there that leave room open for a lot of speculation. Ambiguous nature is not bad by default, but it was handled wrong in this case. The fall being added with little post context kind of limits what is possible and what isn't.
Another change I would make is better pacing. Instead of info dumps at 3 points of the game, the largest of which on disk 2, there should have been more consistent explanation with decent hints being strewn about. The hunt for the Dragon Gods is a perfect example where the plot just stops, just to have the rest dumped on you later on. That would be a great place to add some context to the overarching plot IMO.
As for a bigger change, maybe a smaller cast with more character development. Focus on Guile having a stronger connection to Janus aside from the retcon of a retcon added to Chrono Trigger DS. Norris, Fargo, and the Dragoons is all we really needed for the second half (aside from Harle for a brief time) and make Serge, Kid, Glenn, and Guile the main cast for the first half. Have a stronger reunion after Serge gets his body back as well for FK sake!
But aside from that, I would leave it as is and only tie up loose ends that need to be. It was the first Chrono game I ever played, and led me to Chrono Trigger. Even as is, it holds a special place in my heart.