For short photoshoots close to Reykjavik: you can shoot in Reykjavik itself, walk to the water, wander through the volcanic landscapes with moss around, or go towards Kleifarvatn. Without leaving far from the airport, you can shoot in the Blue Lagoon or near it.
Also a cool option near Reykjavik and Keflavik is the active volcano Fagradalsfjall. It's not a short walk, but definitely worth it! By the way, you can also have a ceremony there.
When leaving for one or more days, the most popular options are the South Coast near Vik and the Snæfellsjökull peninsula.
The most popular and easily accessible attractions are concentrated on the coast of Vik: the black beach with Reynisfjara columns, the Dyrhólaey lighthouse with magnificent views, the Skogafoss, Kvernunfoss and Seljanafoss waterfalls (Iceland's most Instagramable waterfalls and insta spots!), the Fjaðrárgljúfur glacier and canyon. The popular Geyser and Gullfoss waterfall are relatively close: they are less suitable for a photoshoot, but they are definitely worth a look during a trip.
A little further from Reykjavik, behind Vik, there is the Jökulsárlón lagoon and Diamond Beach, views of the Vatnajökul glacier and the Svartifoss waterfall, but it is better to plan for two or three days of shooting. By the way, ice caves become accessible and safe in this region in winter.
The Snæfellsjökull peninsula is distinguished by the proximity of all places for photoshoots to each other and the opportunity to make a circular route: after a ceremony in Búðakirkja, you can go in a circle and visit the stone arch near Gatklettur, the Kirkjufellsfoss waterfall overlooking the most recognizable mountain of Iceland - Kirkjufell, black beaches and many other cool spots.
There are 2 more regions that I think are very cool to shoot, but they are less popular due to inaccessibility and more difficult logistics. These are Lake Mývatn and Westfjords (western fjords).
Around Lake Myvatn there are unusual lava fields, geothermal springs, caves. Nearby is the Selfoss waterfall, where you can drive up from different directions, and as a maximum program - The Arctic Henge (Heimskautsgerðið) in the very north.
The western fjords are very beautiful in general, without being tied to specific places, but there are also 2 super-locations: the Dynjandi waterfall and the best vantage point overlooking puffins, Látrabjarg. There you can come close to the puffins and take pictures with them.
Details of all locations with sample photos from there can be viewed on my map, which I open after booking a photoshoot.