I have a horrible, awful, embarrassing confession. All my sites, including this blog, are run off of super-cheap shared hosting accounts at Godaddy (yes, the guys with the juvenile commercials). For years I think they did a decent job and my sites were not that busy, so it was no problem. But as with most large, cheap hosting companies, they seem to be cramming more and more domains on each shared server. Someone on this server is chewing up a lot of CPU cycles and it's time to move on.
I have switched to a virtual private server account at a new hosting company, as a sort of stepping stone potentially to a dedicated server (my business and I have over 30 web sites so it probably can be justified). The VPS account is cheaper and lets me start learning some new things about managing hosting (e.g. I have access to the root for the first time) but still shields me from some of the server management (e.g. OS updates). And it's cheaper than a dedicated server, so we will see how it goes.
At some point, not quite yet, the site will have some down time when I do the migration. Not sure yet when that will be -- the wordpress database for this site is over 50mb which exceeds the import file size allowed in my data base tools (phpmyadmin for mysql). I have read there is another way to do it, I just have to do some research and tests first. I probably will have to learn to work the data base from the command line.