Install another WordPress site

I have one WordPress website now, and I want to build another site. Do I need to start all over from beginning by downloading and installing the next site and the theme, even if I want to use the existing version and the theme? I want to build several sites, which have different domain names, but will have the same theme and will be linked together to look like one site.
Can you recommend a hosting company, that will be able to help to install the site, or have easy tools for self installation.
Any advice would be greatly appreciated.

Aapinen