Start at Hallgrímskirkja, arriving around 9:00–9:30 for a quieter look at its soaring concrete architecture and the view from the tower. The church itself is generally open from about 10:00, with tower access usually beginning later in the morning; check the day’s schedule before you go, as services can affect entry. Allow an hour, and walk down Skólavörðustígur afterward toward The Settlement Exhibition—roughly 15 minutes through the colorful heart of Miðborg. This underground archaeological site, built around a Viking-age longhouse, takes about 1½ hours and is a good rainy-day stop; admission is typically around 3,000–4,000 ISK, with opening hours commonly running late morning into early evening.
For lunch, continue to Reykjavík Food Hall in Miðborg. Give yourself about an hour to choose between Icelandic seafood, burgers, noodles, and other casual options; budget approximately 3,500–6,000 ISK per person. From there, walk toward Harpa Concert Hall in around 10–15 minutes, passing the cafés and small shops along the way. Spend an hour admiring the geometric glass façade from both outside and inside, and check the foyer and box-office schedule for any exhibition or rehearsal access—much of the public interior is free, though performances require tickets.
Walk east along the waterfront on Sæbraut from Harpa Concert Hall to Sun Voyager, allowing about 20–25 minutes at an easy pace and another 45 minutes for photographs and the bay views. The sculpture is outdoors and free, but the path can be breezy even in August, so bring a light waterproof layer. For the final stop, take a taxi or local bus toward Sky Lagoon in Kópavogur—roughly 15–25 minutes depending on traffic; pre-book an entry slot, especially in summer. Plan on 2–3 unhurried hours for the geothermal water, changing facilities, and the seven-step ritual if included in your ticket. Expect approximately 12,000–18,000 ISK per person; arrive 20–30 minutes early, and keep valuables in the locker rather than carrying them poolside.