Sure, you can take day tours from Reykjavik or book a multi-day tour, but having your own car offers you more flexibility and the freedom to explore at your own pace.
Do not plan your Iceland road trip with the driving times indicated by Google maps. Not that they are wrong. It is just that they do not take into account that you will want to stop everywhere along the way.
Iceland has something to offer every month of the year, so it really depends on what you want to see and how you want to travel. If your timing isn’t already dictated for you, then consider planning your visit for the summer when there is more daylight hours for exploring.
– All the F roads are accessible only to 4WD. Most of them are in the Central Highlands and open only between June and September (roughly as it depends on conditions). Those F roads sometimes have significant river crossings.
– There is already a lot to see not far from the Ring Road and the access gravel roads are manageable with 2WD cars. – I rented a 4WD vehicle and I must admit it was much more comfortable for longer gravel roads. – In winter the roads can be covered in ice and snow.