According to ESRB, this game contains: Animated Violence, Mild Language
Parents should have no worries whatsoever. Aside from one character's penchant for saying "arse" and "bloody hell" all the time, there is nothing in Chrono Cross that I can imagine offending anyone. The game is generally a bright and cheery experience that deals in complex issues but nothing inappropriate for children.
I imagine most people will like Chrono Cross, whether they are familiar with Chrono Trigger or not. It looks good, plays even better, and has a plethora of interesting places to go, characters to meet, and things to do throughout its 30 to 50-plus hours of gameplay.
Those who liked Final Fantasy VII and Final Fantasy VIII will find the look and feel of Chrono Cross both familiar and innovative (incidentally, those of you with an S-Video connection should try it since it does make a visible difference in the textures of this game), while those who didn't enjoy those games as much should find the more open-ended gameplay to be worth a look.
As for fans of Chrono Trigger, I am not sure how they will respond. I was a huge fan of the original and found my reaction to be mixed at points, while I'm sure there are those that will find it excellent from beginning to end. I suppose, it's up for the player to decide. I personally enjoyed the modest implications of Chrono Trigger more than the heavy-handed, pseudo-philosophical overkill than Chrono Cross eventually lapses into, and I would be disappointed if this kind of approach marks a trend the series will continue in.