Always use your best cards, in the same order. (For some reason, this never failed me if I'm winning against that Rare Card weilding person)
You can't play cards with Zell's mom until you are leaving for Timber.
Sometimes it only takes two games, other times it takes 20.

Word of advise:
After you got the Ifrit Card and finished the Dollet mission, challenge the running kid for the Minimog Card in the Garden.

Then the Black guy in the cafeteria in Garden for the Quistis Card.

Challenge Diablo when you first have him while having Siren learned Life Refine Magic to collect Curaga from Tents. Have Ifrit and Thunder Bird learning HP-J. (It'll just be easyer, have Fire/Ice/Thunder Refine Magic helps to cast powerful spells)
Once defeated, you get the Diablo Card.

Go back to the 3rd floor of the Garden, challenge head-master Cid for the Seifer Card and Collect a battle meter if you didn't get it when you became a SeeD (After Squall, Zell, Selphie and Nina got promoted as SeeDs, talk to H-M Cid.)

Zell, Angelo