Lots of people go straight to the guard at the front gate or the Trepie in the cafeteria, but these two are so easy to beat when you only have the starting cards. If you need Elnoyle, Abyss Worm, Tonberry King, Adamantoise, etc. just head to the library as soon as you get your first cards. These two are seriously bad at Triple Triad and drop everything you need in the early game.
Oh heck yes. Diablos has incredible utility! I like to be ready to fight him right away, but even if you fight him well prepared he can just incidentally clap you! Best of all is encounter-none!
Doom train is so tedious to get when you're also trying to stack that one ammo for Irvine. Can't not get him though.