Let me take you around Reykjavik, showing you all the most important spots in the center as well as some hidden gems that only locals know about. I'll also tell you about the history and culture of Iceland. Once you're done, you'll know Reykjavik like a local.
The main church: Hallgrimskirkja, Harpa, The Old Harbor, City Hall, Old Town
In order to start the tour, please make your way to Hallgrimskirkja, the big church in the center. To get there from the center, go up Skolavordustigur street, the street parallel to the main shopping street: Laugavegur.
Places to stop along the way:
You will pass by Harpa, the concert hall. Sea Baron, one of Reykjavik's oldest and best seafood restaurants. I strongly recommend a stop at Babalu café on the way to the church as well. And finally a stop at the local supermarket (Bonus) to buy some skyr, the local Icelandic dessert, rich in protein.
Best time to walk:
Winter: between 12 to 4PM
Summer: Morning or evening are the easiest time
Iceland was rated one of the 5 safest country in the world, therefore you do not have to worry in terms of safety. The only thing I would warn you about is ice: in winter, streets can become very icy, be careful when waking down the streets and if needed, stop at a local grocery store, they all sell small city crampons to wear over your shoes.