In order to obtain every card in the game, you need complete the following side-quests:
the UFO?/Pu-Pu sidequest, the Chocobo sidequest, the CC Club sidequest, and the Queen of Cards sidequest. The UFO sidequest is the only way to get the Pu-Pu card, the Chocobo sidequest gets you the Chicobo card, the CC Club members all sport some decent rare cards, and the Queen of Cards sidequest is the only way to get certain cards. Unfortunately, these side-quests are rather convoluted, and needlessly complex.
Rare Cards:
PuPu - Complete UFO? Quest: encounter UFO at Winhill Bluffs, Mandy Beach (northeast of Timber), Heath Peninsula (island east of Trabia Garden), Kashkabald Desert. Encounter and destroy UFO at Grandidi Forest. Encounter PuPu at the crater of Balamb Garden and give it 5 Elixirs.
Chubby Chocobo - part of queen of cards quest, lose Chicobo to QoC, in order to be able to win Chubby Chocobo from the guy who occasionally sits on the bench outside the library in Balamb Garden.
Angelo - win from Watts in either Timber during the Forest Owls mission on the train, or on the White SeeD ship.
Gilgamesh - part of the CC-Group quest. Win from CC-King, Quistis.
MiniMog - win from jogger kid in blue running around Balamb Garden
Chicobo - complete Chocobo forest quest.
Quezacotl - win from Mayor Dobe at Fisherman’s Horizon.
Shiva - get from Zone on board the white SeeD ship, if you give him the Girl Next Door magazine for free (this can be found in the Timber Maniacs building in Timber).
Ifrit - defeat Ifrit as part of the main quest in the fire cavern.
Siren - win from pub owner in Dollet in his secret room. Challenge and beat him normally, and he will invite you to his secret room. Siren will the be available to be won.
Sacred - defeat Sacred and Minotaur in the Tomb of the Unknown King.
Minotaur - defeat Sacred and Minotaur in the Tomb of the Unknown King.
Carbuncle - part of the CC-Group quest. Win from CC-Queen Heart, Xu.
Diablos - defeat Diablos by using the Magical Lamp item Cid gives you upon leaving for Timber.
Leviathan - part of the CC-Group quest. Won from CC-Joker in Balamb Garden training centre.
Odin - defeat Odin in the Centra ruins.
Pandemona - win from Hotel Owner in Balamb Garden once you have defeated Raijin and Fujin during Balamb’s occupation.
Cerberus - defeat Cerberus as part of the main quest, in Galbadia Garden, during the battle of the gardens.
Alexander - win from Piet in the lunar base.
Phoenix - part of the QoC quest. Lose Doomtrain to QoC and Phoenix becomes available to be won from the presidential aide in Esthar, in the room you drop off Rinoa as part of the main quest.
Bahamut - defeat Bahamut in the Deep Sea Research Centre.
Doomtrain - part of the QoC quest. Lose Alexander to QoC and this becomes available to be won from the pub owner in Timber.
Eden - defeat Ultima Weapon at the bottom of the Deep Sea Research Centre.
Ward - win from Dr. Odine in Esthar.
Kiros - lose MiniMog to QoC, and this becomes available to be won from the man in black in Deling City.
Laguna - win from Ellone on the lunar base.
Selphie - win from Selphie’s friend at Trabia Garden.
Quistis - win from one of the Trepies at Balamb Garden.
Irvine - part of the QoC quest. Lose the Sacred card to QoC, and this becomes available to be won from Flo in Fisherman’s Horizon.
Zell - win from Ma Dincht in Balamb. Only available if Zell is in the party.
Rinoa - lose Ifrit to General Carraway in order for this card to become available to be won from General Carraway.
Edea - win from Edea after the battle of the gardens.
Seifer - win from Cid.
Squall - win from Laguna.
The following guide, however, details the best way to approach getting every card (all credit goes to Sister at GameFAQs):
http://www.gamefaqs.com/ps/197343-final-fantasy-viii/faqs/10...