i am wondering where is that key which opens the rest of the cave in besaid the woman at the store said a traveller bought it where is he as soon as i get it i know what to do

In a mission in Ch.3, a guy you rescue will give it to you.

which dude where in chpter 3, is it the people you save from yojimbo in the cavern of the stolen faythe?

Uhm... was the last post meant as a question? It would be easier to tell would you use punctuation... :(
If it was a question, the answer is, yes, the mission when you rescue people from Yojimbo. I have no idea which guy it was, though.

I don't remember which person it is either, but I do know that it's one of the more obvious ones that you'll probably find easy and is needed just to unlock Yojimbo.

BTW, the reward for unlocking that cave isn't exciting at all. It's just a Garment Grid that doesn't do anything particularly special.

i know you get a search sphere from the chest you open with the key but can the search sphere search for anything else in the game, cept the cyphers hidden in besaid

(im exploring new colours)

No, honestly, the only thing that the Search Sphere is good for is finding the hidden Ciphers and unlocking that secret path in the cave in Besaid, which only has one treasure chest with some Ethers and a Garment Grid at the very end of the outside path. Waste of time really, unless you're intent on collecting all the Garment Grids.