In Reykjavik you can walk most places. The centre is fairly compact.
My main recommendation would be Stofan ... cafe ground floor, very cosy craft ale bar in the basement. Good for people watching in the evening from the comfort of a sofa, if you are in there early enough.
Geysir, Thingvellir and Gulfoss are the Golden Circle attractions. Allow most time to really explore Thingvellir ... some amazing clear water channels in the rifts. Basically you are walking through earthquakes ... between two continental plates.
As Grindavik is closed, you get to the south coast via a more easterly road, which takes you past Krysuvik. Well worth a stop to wander around the geothermal springs.
South coast there's Vik, Skogafoss, take the road up to the edge of Myrdosjokull to stand on the glacier.
A bit further on there's Jokulsarlon .... where the glacial meltwater and icebergs travel towards the sea. There is accommodation in Vik if needed (like a hostell if I remember rightly).
As mentioned above, Sky Lagoon ( a fairly new place) is way better than Blue L:agoon, which is shut anyway.
Lastly, in Reykjavik, there's Perlan .... a fancy restaurant on top of awater tower that offers views over the city... and there's the rocket shaped church of course.