Islands that are easy to navigate without a car?

I’m traveling to Greece from mid-August to mid-September this year and I’d really like to visit an island or two during my stay, but I won’t be able to drive.

Any recommendations on islands that are still good without a car? I’m happy to walk, hire a bike, catch buses etc. I also would love to be somewhere with plenty of stuff to do during the day as I’m not really a fan of lounging - I’d like to be able to go see sites, explore shops, maybe go for a bike ride/hike/snorkel etc.