Downtown Reykjavik is really quite walkable. And should your feet fail you, the public bus system can pick up the slack.
Assuming you are staying within the downtown area… Laugavegur Street, Skolavordistigur Street (known now as Rainbow Street), the Hallgrimskirkja and the Tjornin will be the most important place names to know to keep one’s inner map calibrated.
Walking around the Tjornin - the lake just south of the parliament building - you will find both picturesque photo spots, and a series of interesting sculptures. (I am a particular fan of “The Unknown Bureaucrat”.) Then just west of the Tjornin, and up the hill, one can stroll through the Holavallagardur - a historical cemetery with keenly-decorated plots. Because of those green decorations, this is a cemetery unlike any other you will have ever visited. As you work your way through, notice the difference in the types of trees planted by each headstone. The “in-style” tree seems to have changed every few decades.
There’s a nice stretch of greenway connecting the new tesseract-looking Harpa Concert Hall to the famous Sun Voyager scultpure and the Hofdi Lighthouse, along the waterfront north of Skulagata Street. If you are in Reykjavik to catch a cruise, you may get a glimpse of your vessel from this stretch of coastline.
Heading east from the Tjornin, you'll likely follow the mostly-pedestrian Laugavegur Street. Then splitting off from Laugavegur Street onto Rainbow Street, walk to the top of the hill to the Leif Ericson statue and the famous basalt-shaped Hallgrimskirkja. Take the lift to the very top of the bell tower for amazing views and a proper understanding of Reykjavik’s layout. Then peek towards the Reykjanes Peninsula to see if you can view any of the recent volcanic activity that may be ongoing.
Right next door to the Hallgrimskirkja, then, explore the grounds of the Einar Jonsson Museum. You’ll be moved by some truly amazing sculptures from Iceland’s most famous sculptor.
If you really want to stretch your legs, and have a chance to see the “real” Reykjavik, it is possible to walk from downtown to any of the landmarks mentioned in the next post - the Perlan Museum, Sky Lagoon, Botanical Gardens, Sundhollin, etc. Or walk one way and use the clean and logical public bus system in the other direction.