in FF6, Strago plays the blue mage role with his 'lores'. there's plenty guides on that (like here
in FF8, Quistis learns blue magic for her limit breaks. ff8 is quirky, so it works a bit different here, and she learns blue magic by using items, not by fighting enemies. but the main principles still stand, and she still gets the role staples like big guard, aqua breath or w/e. here
's a list.
in FF9, Quina takes the role. s/he learns blue magics by eating enemies, and here
's a guide on that.
in FF10, Kimahri learns blue magics from the enemies by using his lancet skill, and can then use them for his overdrives. they're called Ronso rages in this game, and here
's a nice guide about this.
in FF10-2 you get back a dedicated blue mage role (gun mage) for all party members. so it's back to the classic approach, and works just like you'd remember from the earlier games. i.e. a gun mage gets hit with a learnable enemy skill, and she learns it for her skillset. here
's a guide.
the later installments have a blue mage role too, but you didn't ask about these, so there ya go. but they're still using the role and experimenting with it. even in FFXIV, playing a blue mage is lotsa fun too. and no, i don't think a blue mage is a waste of time, it's generally pretty fun to play.
as for my favourite blue magic, it's white wind i guess, just because it's often the best available healing spell, if a bit situational.