In all the games, is it the same Link, and Zelda, and everybody else who appear again, or are they different peoples.

It really depends on what game you're playing. There are different Links and Zeldas. Like the Link in Ocarina of Time is different than the Link in Wind Waker, since those two games aren't even in the same time line. But the Link in Majora's Mask is the same Link in Ocarina of Time since Majora's Mask takes place directly after OoT. So basically, unless it's a direct sequel to a previous game, it's not the same Link.

What Saami said, basically.

For each adventure there is a different Link and Zelda except in the following:
OOT!Link & Zelda = MM!Link & Zelda
TWW!Link & Zelda = PH!Link & Zelda

Don't forget LoZ Link = AoL Link ( though oddly enough LoZ Zelda is not the Zelda in AoL )

ALTTP and LA too ( though Zelda's only mentionned in LA, unless you count Marin as based off ALTTP Zelda )

Oracles Link & Zelda are the same as well.

Tanks. I wasn't positive as to whether or not they were de same. If they're not the same though, is it the same hyrule. I mean, in TP when the spirit says this is the clothes of the hero, do they mean a link from one of the past games?

I think it is the same Hyrule. Well not in Wind Waker/Phantom Hourglass, as the Gods flooded it. And all that's left are islands. Then there are the games that don't take place in Hyrule at all, such as Majora's Mask which happens in Termina and Link's Awakening that happens on Koholint Island. (I also think the Oracle games happen in different lands, but I'm not sure as I haven't played them.)

And it's generally believed that the previous hero discussed in TP is the same Link from Ocarina of Time. TP takes place 100 or so years after Ocarina of Time. So I think it's pretty safe to say that The Hero's Clothes in Twilight Princess belonged to Ocarina of Time Link, even though it's not explicitly stated.

So wait. Link in TP got those same clothes, without wrinkles, stains and holes after a 100 year wait?!!! OMG!!!! The goddesses must really like whoever is wears those clothes. Right?

Actually, it was never confirmed that TLoZ!Link/Zelda = AoL!Link/Zelda. That was just a rumor created by suspicious fans who linked the two games based on clothing/character designs. Same goes for ALttP!Link/Zelda being LA!Link/Zelda.

But you are right on the Oracle of Ages!Link being the Oracle of Seasons!Link.

Basically, there are a lot of hints at some Links being the same as in other games, but so far, the only CONFIRMED & OFFICIAL ones are in the Oracle games, TWW/PH, and OoT/MM (and also TP/Link's Crossbow Training.... but that's not really a game. DX )

I always believed that Link's Awakening was a sequel to A Link to The Past. Was that just a rumor?

Just a rumor. According to Miyamoto, "...it's not very clear where Link's Awakening fits in--it could be anytime after Ocarina of Time".

So far, it really has no place. A Link to the Past is supposedly the last game in the timeline (again, according to Miyamoto), but that's likely to change now that Aonuma is concentrating on the timeline. So, Link's Awakening can't be a sequal to ALttP... unless Miyamoto changes his mind. Which he probably will sometime down the road.

Also, if you think 'bout it, the sages in TP are Saria and the rest tfrom OoT. I wonder which one of them dies. Any ideas or theories.

Here's where I disagree. The sages from TP are not Saria, Nabooru, Ruto, etc. The sages in TP are from OoT!Child Link's timeline, and in that side of the timeline, he had not awakened Saria, Nabooru, Ruto, etc. to be sages. So, either they are the sages who were there previously (Rauru being one of them... he wasn't awakened in the adult side of the timeline- he was one of the ancients).... and as we know, there WERE sages before the ones Link awakened, as Kaepora Gaebora was a reincarnation of one, and sages also forged the Master Sword.

So, we do not know who the sages in TP were, but we can guess that Rauru may be one of them.

I read somewhere that the one who died was the Sage of Water (because of the symbol of their pedestal) but I think by the end of the game, they were revived anyway :D