What's the must-see thing in Reykjavik?

Hallgrímskirkja sits at the top of Skólavörðustígur, a sloped pedestrian street lined with wool shops and small galleries. The church took 41 years to build before its completion in 1986, and the basalt-column facade looks like a frozen waterfall from 200 metres away. Inside, the nave is bare concrete, cool even in July, and the Klais organ fills the space with a low resonance you feel in your sternum during weekend recitals. The tower elevator takes you to a viewing deck where the wind hits your face hard enough to make your eyes water. On a clear day you can see Snæfellsjökull, 120 kilometres northwest. On an overcast September morning like today, with temperatures around 10°C, you'll still get the full city grid below and the sharp smell of salt off the bay. The plaza out front fills with coaches by 11am. Arrive at opening and you might have the deck to yourself for 5 minutes.

Perlan, the glass-domed building on Öskjuhlíð hill south of downtown, opened in 1991 and has been reworked into something between a science center and a theme park. The indoor ice cave, kept at -10°C year-round, is the draw. You walk through a 100-metre tunnel carved from 350 tons of compressed snow, and the cold is immediate, sharp enough to feel through a wool sweater. The planetarium upstairs runs a 20-minute northern-lights simulation that is likely better than chasing the real thing on a cloudy night, which in Reykjavík is most nights from September through April. Full admission runs about 4,490 ISK, roughly $37 USD. The café on the observation deck serves a lamb soup for around 2,800 ISK that is better than most downtown options at twice the price. The 15-minute walk up from Nauthólsvík beach is steep but cuts out the taxi fare.

The National Museum of Iceland on Suðurgata has operated since 1863, which makes it one of the oldest cultural institutions in the country. The permanent exhibition traces 1,200 years from settlement-era longhouses to the 2008 financial collapse, and the Viking-age artifacts on the ground floor, including a bronze Thor figurine from the 10th century, are worth the 2,500 ISK entry alone. The building is quiet. You might share the upper galleries with three or four other visitors on a weekday afternoon. That stillness is part of the appeal. To be fair, the Reykjavík Art Museum at Hafnarhús on Tryggvagata pulls bigger crowds for its rotating Erró collection, and some visitors find it more engaging. But for first-timers trying to understand Iceland rather than consume it, the National Museum gives you the clearest 90-minute orientation to the country you've landed in.

A few things first-timers tend to get wrong. The Blue Lagoon is 47 kilometres from Reykjavík, near Keflavík airport, not in the city. Book it on your arrival or departure day, not as a day trip from downtown. Laugardalslaug, the municipal thermal pool at Sundlaugarvegur 30, costs 1,150 ISK and gives you the same geothermal soak the locals prefer, with hot pots at 42°C that smell faintly of sulfur and feel like climbing into a warm bath on a cold morning. Skip Sun Voyager, the steel sculpture on Sæbraut. It photographs well but takes 90 seconds to see. Reykjavík scores 9.3/10 for solo-traveler safety (see the TTDI Solo-Safety Index), and the compact 2-kilometre downtown grid between Hlemmur and Harpa means you can walk everything on this list in a single afternoon without a bus or car.