Blue Lagoon;
Whale Watching and Museum, Husavik;
jokulsarlon;

Blue Lagoon, Perlan (good museum downstairs, 360 degree view of Reykjavik on the observation deck, great ice cream….but expensive food…), The Golden Circle (Gullfoss, Geysir, Skogafoss—and you’ve got to climb Skogafoss…jump up and down and feel how hollow it is up there!!!!) and Thingvellir for Icelandic history and the Mid-atlantic ridge (where the North American and European tetonic plates are slowly separating)
The pic is the view from the top of Skogafoss.

the golden circle tour is a must and its compact if you’re only there for 4 days. Jökulsárlón is amazing if you can see it.

If you’re renting a car, I would really recommend Jökulsárlón glacier – it’s really amazing and the drive takes you through lava fields and through the immense flood devistation that happened a couple of years ago when a volcano erupted underneath the glacier. Also, Thingvellir is really interesting as well as the other stops on the golden circle – gullfoss and geiser are a must!. If you just drive around, you’ll see that most, if not all, of the cultural/panoramic stops are pretty well marked.

if you love nature, dont spend too much time in Reykjavik (I was in rey. just one day and spent the rest of the week to explore the south Iceland instead, and it was totally amazing). I agree with the others, is a must + Gullfoss and Geiser too! It would be a bit tight if you want to make it to Jökulsárlón when you have only four days, but its totally worthing!

Akuary, (please don’t mind my spelling) I would concern myself less with where to go, because the prior suggestions are great, but concern yourself with when and what. If a festival is scheduled on the island find it. if there is going to be a large gathering of drunk icelander’s join them. The people make the island not the places. with that keep with the gold circle and Reykavik is always a catch all.

I would definitely recommend the Blue Lagoon!

If you like hiking, try the hiking trail from Landmanalaugar south to the coast. 4 days if you like to take it easy, 2 days if you are a pro. Take good equipment with you, the weather is ugly at times.