The first website was launched on 6th August 1991 by Tim Berners-Lee at CERN. Websites were text-only to begin with, but it wasn’t long before they started to use graphics. The Web Design Museum collects examples of designs from 1995 onwards.
You can browse by year, category, and style, as well as looking at timelines to see how the design of well-known websites such as Apple have changed over the years.
The Web Design Museum doesn’t have any museum websites yet, so I took a look at one example in the Internet Archive: the British Museum. Its site started at an academic domain (british-museum.ac.uk, now defunct) before moving to its current address.
See also: Can a website be a museum?