I remember stopping in the middle of the original campaign back in the day as I wasn't impressed with it, but reviews say the expansions are where the single player shines.

Also, I tried googling, but I guess I didn't know how to phrase my search properly. Sorry.

But! Having both expansions installed improves the original campaign playthrough a lot, IMHO (items, feats, classes, etc.).

Sure, D&D3.0 is not very good. The expansions help the game in getting the adapted D&D better, even in the base campaign (but still it is not as good as D&D3.5 from NWN2).

Shadows of Undrendtide is designed for you to begin with a fresh level 1 character. You certainly could take your character from the original campaign into the next one, but it's not recommended.

Hordes of the Underdark is technically a continuation of Shadows of Undrendtide, but since only one NPC makes an appearance in both campaigns this can be easily ignored. Aside from the fact that you need to begin around level 14, there's nothing stopping you from just jumping straight in with a fresh character.

Is there some linkage between OC and Shadows of Undrendtide?
Or a completely different story?

Completely different story. There are no characters who appear both in the OC and in SoU.

Hordes of the Underdark is a little strange. Your character is the same one from SoU, but you actually meet up with many of the characters from the OC, so it serves to tie together both campaigns nicely. A character can jump into HotU and pretend it's a continuation of the OC with minimal hand-waving.

The only connections are OC and Hordes of the Underdark and also Shadows of Undrentide and Hordes of the Underdark.

You could more or less skip Shadows of Undrentide if you want since you can build a new character in Hordes of the Underdark which get boosted to about level 14, but a completionist would play Shadows of Undrentide anyway.

That say, you SHOULD play the Original Campaign and Hordes of the Underdark. Fantastic games. :P