I did my research, waded through loads of reviews, got completely baffled as I trawled through all sorts of technical descriptions and then I bought this one, a Sony W800:
photo from http://www.sony.co.uk
It was like my old Kodak, in that it was small and would fit in my pocket. However, once I had charged it up and tried it, I found that there was a delay between clicking the shutter button and it actually taking a photo. This wasn't great as I do like taking the odd photo of wildlife now and again. I also found it took a while to bring the photos into focus on the screen. Whether this was a problem with just this particular camera, or across the range, I wasn't sure, but I knew it wasn't the one for me, so took it back.
I have now bought this one (another Kodak - FZ201 - but a little bit bigger and with more zoom than my previous one).
image from https://www.ephotozine.com/
I have been having a practise with it in the garden...
Finally, my replacement hanging basket, still flowering and looking good (until the first frosts come, which can't be too far away now).
So far, I am really pleased with my new camera. I hope it lasts for a good few years.