What I really wanted to use was Adobe Myriad Pro for the headings and Adobe Minion Pro for the body, and I was fully prepared to buy a one-off perpetual licence to use them as webfonts. However, it seems that Adobe now only licences these as webfonts when used through Adobe Fonts, which is a great shame. If anyone knows where I can buy a licence for them, do let me know.
I toyed with all sorts of other alternatives, but in the end I went for the IBM Plex family of fonts. I’m using Plex Sans for the headings, Plex Serif for the body text, and Plex Mono for any monospaced text. I think they look quite clean and readable, but friendly with a bit of personality. They also happen to be free and open source, but that’s a nice bonus. I would be prepared to pay a reasonable price for a licence to use them anyway. I will probably do a bit more tweaking along the way (I am still figuring out the best way to specify and load the fonts to keep page rendering snappy), but for the moment I am happy with it. I haven’t changed the design of the site for a few years, so it might take a me a little while to get used to the change in appearance.