As for types I hate facing on opponent teams:
Dragon: the four main elemental types (Fire, Water, Grass, Electric) are crap against the Dragon-type, and Dragon Pokemon usually come packed with strength, a wide movepool, and ferocity to decimate entire teams. Ice-type or Dragon-type moves are frequently needed to handily dispatch of these guys - with other offensive types, it's going to be a very rough ride trying to take down these guys in a fight. Compounded to all of that is that most of these guys are "pseudo-legendaries", which means "absurdly high power stats", worsening the problem of combating them even more.
Ice: the primary vice to my Hasty Flygon, KOing him in one hit. Usually the Pokemon doesn't even need STAB or respectable Special Attack. :(
Steel: if you don't have Fire, Fighting, or Ground type moves, they are a pain to take down, because they are resistant to NEARLY EVERYTHING.
Water: They tend to have high Special Attack, which when coupled with STAB will definitely hurt even at neutral damage. Not only can they drench your Fire-types and have few offensive resistances, they also have the capacity to use annoying Ice-type moves which can typically KO without STAB needed.
Psychic: can raise their Special Attack to absurd levels after they bother you with status effects and other techniques, subsequently pwning you in mere turns after they set up. And several of these moves have annoying secondary stat drops.
Electric: Paralysis. Paralysis. Paralysis. Usually very fast so you'll end up getting paralyzed on their first turn. In games like Super Smash Brothers they're even worse, because you can get trapped in devastating electrical combos, get stunned, and can barely ever stage a decent counterattack or escape from said combo. While stunned, you get beat up yet again, and the cycle of electrical torture continues.
Normal: something is always funky with these types, whether it's high Defense, Special Defense, HP, or a frustrating ability.
Ghost: very annoying secondary effects, tend to have Levitate, and can set up punishing combos.