Before thinking about designing a mobile website, let’s talk about website desktop version first. The last 15 years web designing has changed, so did we. With improvement of technology, new ideas and approaches have taken place.
With CMS (Content Management Systems) such as Joomla, WordPress, among others, designing became rare.
Today is more cost effective buying a theme and customizing it. But don’t think it’s an easy job. Customization takes time, it’s the beginning of branding. It needs to be done right!
Have that been said, most themes come Mobile Responsive which means the website will self adjust based on the screen size of the user. There are also tools to customize a version for mobile or tablet if preferred.