ufo: you should use the ability to prevent random fights, it makes the searching a lot easier.
first you have to find the ufo in 4 places:
monday-coast in the south of timber
around winhill
kashukbaar-desert (desert on the southern continent), in the western part of the desert
island north-east of the heades-coast (continent of trabia)
the screen changes to battle-screen, but you can't attack the ufo, just enjoy the sight!
when you have those four points, you have to fly with ragnarok to the north of esthar, on those strange mountains in the north of the continent. get out of your ship and search for the ufo again.
fight it, it's very easy
now go to balamb (make shure to have the ability ITEM and some elixiers in your inventory), around the place where balamb garden was.
when you meet the alien, give him some elixiers and you'll recieve the card.