Help with Iceland trip plan

First suggestion--get a good Guidebook. (Rick Steves has one.) It will answer all your questions.

You could only explore about one fifth of the country with day tours from Reykjavic. If you want to experience the country you need to move about.

When I went to Iceland about 6 years ago now, it was difficult to find a guidebook. I ended up with a DK Top 10 book. Now RS has one and its pretty good. Because I went alone, I stayed entirely in Reykjavik and took a few tours and walked/used the city bus. You might stay in Reykjavik a few days without a car and do some exploring and then pick up a car and explore the island a bit. I don't think you need a car for Reykjavik, itself. Accommodations are expensive. My DIL used AirBnB a lot.

There are a lot of great things to see that can be done as day tours from Reykjavik, but with 9 days I think you will run out. I spent 5 nights in Reykjavik a few years ago. I did the Golden Circle one day, Thorsmork another day, Snaefelsness peninsula another day, and that still left me a full day to explore Reykjavik and time to visit the Blue Lagoon on the day of my flight home.

I agree with Lane that with that length of time, you'd run out things to do in Reyjkavik and will want a car for further exploring.

I agree that you should rent a car and use two to three cities as a base throughout your trip. The costs of the day tours for that many days will be about the same as renting a car but then you get the added benefit of being on your own itinerary. My picks for bases would be the south coast (maybe around Vik), Reykjavik, and the Snæfellsnes peninsula. You could also just spend more time in the south and use a day to go out to the westman islands.