Web sites are more than just their template. You also need to consider hosting costs, domain name registration costs, secure server certificates, and more. One assumes that you have at least some of that covered and are at the design phase.
You may want to explore using WordPress to run your site. You can set up a site at wordpress.com and pay just a little extra per month to have your own domain name point to the site or you can download WordPress and host it yourself if you have web hosting already set up. Shared web hosting that allows PHP and MySQL is common and fairly cheap to obtain. In either case, there are many free themes available either included with wordpress.com hosting or via the WordPress.org site if you self-host and a lot of them are built on HTML5 and will provide a good user experience.
Whether you host a WordPress site or choose to go with a different CMS or no CMS, please be careful in sourcing free templates and only download from known sources. A great many free templates are infected with malware or contain other forms of security holes that allow a malicious operator to hijack your site.