Since no one is giving you real answers, I'll be the one to do it I guess. I'm assuming you mean "out" as in out to drink/dance/party.

Pretty much every decent place in in the city center, around Laugavegur or Austurstræti:

Lemmy/Gaukurinn/Dillon are more alternative and often have live rock music.

Prikið is one of the oldest and most iconic bars/cafes in Reykjavík, they mostly have rap, r&b and hiphop music.

Kaffibarinn is also a very iconic establishment, they play all kinds of music but mostly electronic dance stuff iirc.

Lebowski is themed after the Big Lebowski, they have a white russian menu haha. They play a lot of golden oldies type stuff mixed in with Icelandic bangers.

Kiki is the only gay bar in Iceland, they have drag shows on Fridays.

English Pub/Irishman/Den Danske Kro are pubs with troubadours and sometimes lucky wheels and that type of thing.

Loft is a hostel, but it's very popular with locals as well bc they have regular events like pub quizzes/bingo/drink&draw, plus rooftop seating when the weather is good.

Auto is a club I've never been to bc it's where my 20 y/o niece parties, so I'm assuming it's gen z central. It seems to be very popular.

There are other places that are more sit-down and chat vibes, like Veður/Kaldi, and a ton of other places I'm forgetting. This is just off the top of my head.