How do you do the "wavy" vocal sequence like an opera singer? You know, it's like a-h-a-h-a. I'm not sure how to describe it, but it's kinda like when an opera singer or a singer in general, such as Celine Dion or Britney Houston, makes a long "wavy" vocal sequence, especially at the end of a verse.
You might be disappointed in this answer, because it's nothing special! Based on the link you gave in a comment, this is just something called a melisma, which is just a string of pitches sung on a single syllable.
Think about singing a children's song or a hymn; most new syllables are a change of pitch or a re-articulation of the current pitch. (Think "Old McDonald," "Itsy-Bitsy Spider," etc.) Occasionally you'll sing a string of pitches on one syllable, but it's relatively rare.
With a melisma, however, it's one syllable for a string of several pitches, which is all Celine Dion is doing in your example.
Although occasionally a vocalist may switch between an "ah" and an "oh," I'd say this still falls neatly within the melisma tradition.
There's no specific technique to it; just sing it well!
You mean coloratura? Like in the Queen of the Night aria? Well, you do that by having good teachers and practising a whole lot until you have a solid voice support and solid control over it and solid pitch and solid delivery and good breath elasticity and well-rounded leggiero. There is no particular trick to it. You just do everything right and well-controlled and coordinated and it will sound ok on a practised voice.