We were just there in April, but haven't had the opportunity to post a trip report. We stayed at the Icelandair Marina Hotel in Reyjkavik, which was perfectly located. We're glad we didn't pick the Natura or the Hilton. It's in a former paint factory, but remodeled and is the newest of the Icelandair hotels. They're adding a new wing, so part of the hotel was closed off due to construction. The other part's nice (and they have a fantastic breakfast buffet), but the bar downstairs blasts disco music past midnight--especially on Friday, so if you plan to sleep and not join the late-night party, try to get a room that's not above the bar. We were on the 3rd floor and the thumping bass sounds were annoying at best.
Definitely go on a whale watching tour - we picked Elding, the most experienced company. If you don't see whales/dolphins/porpoises, they'll give you a voucher for another free trip that's good for 2 years. They also offer puffin watching tours in the summer and Northern Lights tours in the winter and early spring. Their trips leave just a few steps from the Marina Hotel, and we spotted a minke whale!
Lots of car rental companies have pick-up/drop-off locations at the Keflavik and/or Reykjavik airports, at the central bus terminal, or at locations in or around Reykjavik, and some offer a pick-up service. We used Sixt, which has a location a short walk from the Marina Hotel. If you take the airport bus to/from Keflavik airport, Sixt was easy for collecting the car and turning it back in. Definitely get a 4-wheel drive model. Economy runabouts are much cheaper, but if you go anywhere outside Reykjavik, even if you drive part of the Highway 1 Ring Road, you'll want to take a side road or two, and having extra clearance and 4WD is worth the expense. We drove south one day, north one day, and east for the Golden Circle a 3rd day.
We didn't arrange for riding horses, but that was clearly a popular activity for many people. You can drive to one of many stables or horse farms, or sign up with a tour company that will collect you and drive you to the riding site. We went to the Fontana baths rather than the Blue Lagoon. You pass thru Þingvellir to reach them from Reyjavik.
We signed up with Icelandic Mountain Guides for a 2-day tour that included a visit to Jokulsarlon glarier lagoon, a 1-night stay in southeast Iceland, and a glacier walk with crampons that included passing through a crevasse! They also offer many one-day activities, are very professional, and are highly recommended. We considered visiting the Heimaey in the Westman Islands, but in early April, many sights were on very limited opening days and/or hours, and the puffins wouldn't be there yet. September may be just outside their High Season, too. We also considered visiting the Snaefullsness peninsula, but with our other activities and just one week, saved that for another trip. With just 3 days, you'd have to really want to see Snaefullsness and make that the focus of your visit. The Westfjords, the island of Grimsey (the only part of Iceland actually north of the Arctic Circle), and the rest of northern and eastern Iceland also will have to wait for another time.
Because of Reykjavik's location on the western edge of the country, many tours that go to destinations elsewhere then deposit you back in Reykjavik spend a lot of time on the road, in a bus or van. Driving yourself may work better, or may not.
One consideration, though, there's a joke that if you don't like the weather in Colorado, wait 10 minutes, and it'll change to something different. In Iceland, it's more like 10 seconds--we saw sun change suddenly to sideways sleet, then calm, then fierce wind. Driving Hwy 35 near Selfoss on a dry road at 32°F, a VW coming towards us suddenly flew off the road, flipping 5-6 times. Black ice had suddenly formed just ahead. We stopped, called "112" and tried to help. Two people were thrown from the car, one died. Buckle up!