The fighting is slow compared to most other FF games, but if you put the Battle Speed to the maximum, it really isn't that noticable. It hasn't affected my enjoyment of the game one bit in all the times I've played it.
In terms of the story, it isn't slow at all. The story picks up quickly after Alexandria, and especially after Lindblum, and it's pretty much non-stop from there on. It may be a tiny bit slow right at the start in Alexandria, when you're walking around with Vivi, but it's just setting the scene then, and it's such a small part of the game anyway.