I got into the series with FF6, released as FF3 in the states. While I enjoyed FF7 I thought it was a step down, but great nonetheless. I don't get why anyone would want that game remade, though. Square should just release for PSN. But knowing Square that will never happen as they love milking their franchises and making you rebuy your games again and again. Man, their getting in Nintendo territory in that regard.
The big weaknesses that I remember at the time:
Localization. I still don't understand a lot of the stuff and the end of Disc two.
Sephiroth. What a spolied brat. Amazing boss theme though. But nothing compared to Kefka in terms of evil
Character Design. Don't get me started with that. I liked his FFVIII and FFX designs, though.
Materia. I spent way to much time mastering materia. In the end I had complete Master Materia sets for 4 characters in all the main materia categories. But I maxed out the game clock and wasted so much time.
Characters. In the top levels the characters, with the exceptions of their limit breaks, were virtually indistinguishable from each other again due to the materia system.
Again, while its raving fans could be annoying, I'm so glad it was such a big hit as that meant more localized JRPGs for the U.S., and it also introduced many of gamer to what you would call a niche genre previously.