OP credits = Elfen Lied -Lilium. Perfectly sets the mood for basically any of the fourteen episodes.
Actual opening of an anime = Dragon Ball. Entire first episode. Stock poser answer, but it just captures the VIBE of the entire series so well. Colorful, humorous, establishes characters that are still in people's minds today, and establishes the traits of those characters very well within 22 minutes. Has some mystery, sets up a quest, has two unlikely allies team up. Just ah. I can't say enough good things about the first episode (or arc. Or series, if we're being honest) of Dragon Ball. I kinda wish Toriyama went with his original plan and had the series finish up with the Frieza arc. It's puts a nice bow on Goku's journey and in a vacuum without knowledge of the Cells and Beeruses and what have you, Frieza makes sense as a final villain/final exam for everything Goku has been developing his character arc as the stereotypical hero of the heroes journey for. But that's a whole tangent and I took some gabapentin before typing so...yeah. Dragon Ball. First episode.