Rent a car
Take a day or three to get out of the city. My favorite moments were the ones I spent outside the city. The landscape is so beautiful. It's nice to be able to see it on your own time vs. going on a tour.
Get the Lonely Planet Iceland Book
Again, you can take mine :) This was really helpful in finding fun adventures along the way.
Get mobile wifi
You can take mine :) This will help you navigate big time. Load where you want to visit while you have wifi access and then drive! That way if you lose signal, it is already navigating.
Get your camera ready
You will take over a hundred pictures in like an hour. If you need to borrow my camera, let me know. But take extra batteries/storage/etc.
For a 4 Night Stay:
I recommend longer, but this is packed in. Staying longer will let you spread it out and maybe even see a little more in the northeast.
Explore Reykjavik and catch up on sleep the first night at OK Hotel (Bottom Right image)
I have all my Reykjavik recommendations separated in another section below.
Start your drive towards Vik. Stop at these places listed below:
On your way to Vik in this order: Seljalandsfoss (explore the waterfall and cave to the left. Just keep walking past the waterfall and you'll find it), grab lunch and visit the cows at The Old Cowhouse — Gamla Fjosid before the hike to Seljavallalaug pool. (it takes about 15-20 minutes, but TOTALLY worth it). The pool can be hard to find. But right before the pool is the volcano information center. They have a map to it in case you get lost. Then see the stunning Skógafoss. Spend the night in Vik. If the weather permits and you still have time, explore Vik this day. It's small but insanely beautiful. My favorite place by far. Airbnb has options or you can stay at the hotel there.
Explore Vik — Drive to Gulfoss/Geysir—Spend the night in Stykkisholmur
If you didn't get to yesterday, visit the Vik beaches and spend some time there. Trust me, you'll want to! Visit the beaches in this order and you'll be able to head back towards Reykjavik. This is as far as I would go given your time there. First, Reynisdrangar beach which is located right in the small town. Then drive back toward Reykjavik to Reynisfjara beach. The basalt columns here are out of this world! Then go to Dyrhólaey beach for a quick stop before heading to Gulfoss & Geysir. Once you leave Dyrhólaey, take the trek to Gulfoss & Geysir. These are really great places to see and don't take very long. Then take about a 2.5 hr drive to Stykkisholmur. You'll also pass through picturesque Thingvellir park. You're bound to stop for photos. It's gorgeous.
Explore Stykkisholmur — Drive around the Snaefelsness Peninsula — Spend the night in Reykjavik
Stykkisholmur was my favorite place. It is an old fishing town and it's just so charming. Check out Leir 7 and do a viking boat sushi tour here. No need to research which one to do — there's only 1 at the dock. Stykk is a small but wonderful town. Then take the rest of the day to drive around the Snaefulsness peninsula. Stop at these two places along the way for sure: Hotel Budir (not to stay, but visit the black church next to it—incredible). Arnaestapi is gorgeous. Explore it on foot. There is a rock bridge there that is pretty stunning. The pensinsula is really one of the places where you'll just want to stop and look at everything. The landscape here is unreal. Finish the drive all the way back to Reykjavik and spend the night. It's a little gritty and the floors are not as clean as one would like, but if you don't care, then you may check out the Kex Hostel. If you don't sleep there, I'd definitely recommend it as a place to go at night for music & drinks. It's filled with backpackers/nomads and locals.
Spend the day in Reykjavik OR Blue Lagoon & Head Home
This may be a great day to explore what you did not get to do on your first day in Reykjavik. I imagine this will be the day you leave to go back home. OR go to Blue Lagoon. My mother and I timed our trip to visit Blue Lagoon on our way back to the airport. We could've spent hours there. It's absolutely incredible. I highly recommend this either on your way in or on your way back home. BUT (and we didn't know this) you have to make reservations ahead of time. So once you know your flight details, I would go ahead and do that.