I don't think they are supposed to start out with any fighting skills, unless they had them as humans. There are quite a few examples in the series of newly turned vampires that are plain incompetent. The episode mentioned in the question, "Lessons", spells this out a bit in the bit quoted in the question, and a few lines later, there's also this.
Buffy: He's a vampire, OK? Demon. Preternaturally strong. Skilled with powers no human could possibly ever—
Vampire (trying to get out of his grave): Excuse me. I think I'm stuck.
Their demon blood gives the vampires superhuman speed, strength and some kind of predatory instinct. That makes them dangerous to ordinary humans even if they're just lunging randomly. But a newly turned vampire is no trouble to Buffy, and even the rest of the gang stand a chance against them once they've actually had some training. Experienced vampires (for instance, Darla, The Three, Drusilla, or Spike) are shown as much more of a challenge. If the vampires survive long enough, then they can start learning the best way to fight.
I'm rather thinking that Harmony's continued incompetence demonstrates this - a vampire has to practise to learn to fight, the same as anyone else, and I can't imagine Harmony practising her martial arts skills when there are gossip magazines to be read. I imagine Spike tried to teach her at some point. I imagine he gave up in despair pretty soon.