Hi,
You've asked the same as I did many years ago. As mentioned, you need a domain name and a hosting service. There are many choices and options. These can be obtained from many sources such as noted by WeatherHost, or well known like GoDaddy, etc. Or do it yourself but that will need a bit more knowledge.
Several of the weather programs available, such as Davis Weatherlink and CumulusMX, have built in website files and processes to update a weather website. Once you have a hosting and domain these programs, through some basic setting options, will pretty well do the rest. You'll also need a text editor program, like Notepad++ which is free) to do some file editing or customizing if needed, and an FTP program like FileZilla (also free) to do the basic static files uploads to your webhost. Once you feel comfortable then you can dive into the more elaborate or your own ideas.
You'll find lots of advise and suggestions to your specific questions here.
Enjoy,
Paul