Following this advice, I discovered an extension that automatically spoofs the user agent and does some other things. Haven't tested yet, but it seems to be actively maintained.
Since I'm on the topic, other two lesser known extension I have are CanvasBlocker (fakes canvas fingerprinting (or disables it)) and Privacy Badger (heuristically detects and disables trackers; complements ublock).
You have to stick with known popular user agents. To mitigate tracking by UA you need a randomized user agent that changes periodically. Panopticlick won't be able to account for that in its stats. It's not a good idea to switch UA on every request since it will be hard to diagnose breakage caused by a site that rejects particular UAs.
"Periodically changing user agent" sounds pretty unique if you ask me. Especially if the extension isn't super-clever and changes the user agent for accesses that happen on the same page.
And the fingerprinter could be super clever and look for features that your purported browser isn't supposed to support... And if your browser does support them, that's a strong identifier.
Unless you use some obscure browser, it is better to use your real user agent. If you keep your browser and operating system up to date, chances are it will be one of the most popular ones.
Your UA will correlate with other means of fingerprinting you making you more common. Being clever can make things worse.
For example, the most common UA is from an iPhone, but the most common screen width is 1920 pixel. If you decide to make your UA an iPhone with a 1920 pixel screen, then you will be easily identified.
