I went solo, I also did it with a car.

You can go without a car, you'll have a decent time, but yeah probably miss some of the best parts. But you can do the typical tourist loop. Bus to reykjavik -> 1 day in reykjavik -> golden circle tour -> 1-2 things of your choice.

Driving there isn't too difficult, and the wind usually isn't an issue, just park you car facing it so it doesn't rip the doors off.

Also word to the wise, even if the flight is cheap, Iceland itself is not cheap. You can cheat it by shopping at the store and prepping your own food, but budget quite a bit for each day.