Traditional solo trip?

Hi everyone! I’m thinking about doing a small trip in May for a long weekend (early Friday to Monday or Tuesday) and was wondering where you recommend going to experience a more traditional side of Iceland?

I’ve been to Iceland twice when I was a teenager so I’ve seen the golden circle, I was thinking of flying to Keflavik (from the UK btw) and getting a rental car and maybe driving to the east side while stopping overnight either on the north or south hopefully completing the ring road over the trip.

Cost/expenses are not really an issue but I’d rather not spend more than I have to and I’m happy staying in cheap accommodation. I have no issue long distance driving, I used to drive 6 hours across the uk every week/2 weeks to see my partner who has recently broken up with me so this trip is more of a therapy trip! I wouldn’t think of the drive as a waste because it’s very relaxing for me and I get to see this amazing country

Back to my original question, I would love to see a more traditional side of Iceland (not the big city), I’d love to meet people and I enjoy pubs, bars etc so that would be good to go to. I’ve done a bit of research online but it would be good to hear from people who have been there and experienced it for themselves. So if anyone’s got any recommendations that would be amazing!

(PS I’ve always wanted to visit the east side so anything over there would be great, please let me know if I’m overestimating the drive but honestly I can grind out a big drive if I know what’s on the end is worth it!)