Reykjavik is the natural base for the southwest of Iceland, and a surprising amount of the country's headline scenery is within day-trip range.
The classic loop is the Golden Circle — Þingvellir National Park (where the Eurasian and North American tectonic plates meet, and where the world's first parliament sat), the Geysir geothermal field with the still-erupting Strokkur, and the thundering Gullfoss waterfall. Buses and self-drive both work; the full loop is about 230 km and takes a long day.
Closer in, the Blue Lagoon is forty minutes toward the airport — a milky-blue geothermal spa famous enough that you should book ahead. The newer Sky Lagoon, on the edge of Reykjavik itself, is cheaper and quieter. The Reykjanes peninsula, the volcanic landscape between the city and the airport, is one of the rare places where you can see the Mid-Atlantic Ridge above ground; recent eruptions near Fagradalsfjall have only added to the appeal.
For something further: Vík and the south coast (black sand beaches, glacier tongues, waterfalls) is do-able in a long day; the Snæfellsnes peninsula to the north is a more relaxed twelve-hour loop.