Definitely rent a car and do the Golden Circle drive. I saw some of the most beautiful things I've ever seen on that drive. Reykjavik is an awesome, but expensive city. Be prepared to drop $30-$40 a person on average food. Beer is around $12 a pint. I wanted to buy a couple of souvenirs, but didn't because of the prices. A crappy t-shirt will set you back $40. That being said, I can't wait to go back.
Other things to try:
Icelandic hotdogs (Baejarins Beztu Pylsur is the famous cart, but they can be found many places)
Gull and Viking are the two Icelandic beers. Both are good!
Go see the Viking sculpture on the water.
Check out Hallgrimskirkja church and the Leif Ericsson statue.
Harpa Music Hall
Walk along Laugavegur, the "Main Street" of Reykjavik.
Here is a good guide to the Golden Circle drive:
https://www.google.com/amp/s/expertvagabond.com/golden-circle-iceland/amp/If I go back, I would love to do a puffin watch. We didn't have enough time. Hell, if I go back, I'm doing the entire Ring Road drive.