How long are you going for? In my opinion Reykjavik is kind of boring as a city so I would get out of there as soon as you can. It's worth a day or two to see the sights but beyond that there isn't anything there to do but spend shit loads of money on food and booze. I've only been down south but there are great waterfalls (they are everywhere) and some very cool beaches like Reynisfjara (my favorite spot I went.) The thing to do is just travel around the ring road and stop and see whatever strikes your fancy. Lots of hot springs that are pretty fun to go to.
Lobster soup at the SeaBaron restaurant.
Gotta try the hot dogs
Otherwise do your own grocery shopping because the food there is insanely expensive.
If you're going to rent a car, check out Sadcars. They are super cheap to rent (like 25 euros/day for the cheapest car) and there isn't a whole bunch of bullshit when renting through them.
Blue Lagoon is nice but touristy and expensive, was still fun.
I'll try to think of more stuff when I'm off work, but that's kind of what I remember.