I would like to post photos on a site that is accessible to others but that is A) somewhat respectful of personal information (i.e. nothing owned by Facebook); and B) is not too silly looking (Imgur). Is there any good option, or is Flickr still worthwhile? I don’t mind if the site charges a reasonable fee. Thank you.
Flickr is a great platform to begin with and you can use the Flickr storage as a backend for your own website. It’s not as fast as using an image CDN, but the speed difference is not that noticeable.
Take a look at Nanogallery: https://nanogallery2.nanostudio.org/. With some very basic PHP/HTML skills and understanding of JavaScript you can go a long way.