If you just want to play the main story, there isn't too much grinding needed, but it does help in the long run. Post game though, there is a ton of grinding involved. The lv. cap is 9999, and even then, you'll most likely need to do reincarnating in order to boost your stats. Thankfully though, there are a lot of ways to make the process pretty easy.
Yes you can revisit areas to train. In fact some levels are designed around grinding. For example, in some levels, there are a group of enemies in a 3x3 formation standing on EXP+50% panels. So you can kill them all and go through the level multiple times real quickly and get your characters up to a desired level relatively fast.
Story wise, yes, the games do have a pretty strong focus. They are a bit light hearted and don't take themselves too seriously for the most part, but they also have some serious moments as well.
I notice that 3 and 4 spend more time telling a story compared to 1 and 2, but I like to think it's mainly due to the limitations of the PS2.
And yes, the game is like tactics in which you move characters on a grid. The main difference being ( from what I've seen in tactics anyway as I've never played Tactics)is that you can move all your characters at once rather then by going by turn order. As can your enemy.
Hope this helps!