Hi Belenteja,
I got back from a 2 week trip to Iceland last weekend. It's certainly worth the trip!
We hired a car for the second week, and would recommend as follows:
if you are sticking to the main N1 route around the island, and roads that are marked as red (tarmac) on the road map, then a normal car is perfectly good. However, if you're going to be travelling on gravel roads (not F roads, regular gravel roads, marked in brown on the road map), which you are allowed to do in a regular car, I would recommend getting a small jeep if you are going to be doing lots of distance on gravel roads and don't like being jolted around for mile after mile. It is possible, and included in the car hire agreement, but the suspension of an ordinary car doesn't give a great ride - and it takes a long time to get anywhere!
There's lots of stuff accessible by tarmac and gravel roads, and it's easy to pick up the Icelandic buses for those destinations only accessible by F road / jeep. If the cost (which is steep) of hiring a car / jeep is prohibitive for you, then I would recommend looking into the buses to travel around (if you have a more leisurely pace than we did!)
We went to Geysir and Gulfoss, and Pingellir (the Golden Circle) which involved only a short stretch on gravel road, and the Snaefellsnes penninsula and penninsula area around Rejkyavik is mainly accessible by tarmac road. The NW peninsula area to Latrabjerg was much more difficult in a regular car, and the gravel roads were very hard work.
Certainly the interior of Iceland contains many F roads, and some of them are only accessible by larger jeeps anyway and probably not recommended if you're not experienced in crossing fast-flowing braided rivers (eg the one to Thorsmork) anyway, but we found masses to do and see sticking to the N roads.
NB If you are going to hire a car - recommend the Kortabok (Mapbook) sold in many book and tourist shops in Iceland, though don't bank on it always being right about the location of campsites! You might be able to find it online to help your planning. This book lists the tarmac and gravel roads as red/brown respectively, as per my descriptions.
Hope this all helps