To get started if you are new to this, or to avoid having to do back-end maintenance yourself on your site, the way to ‘go WordPress’ is to create your site on wordpress.com. The site can be built and hosted entirely free if you don’t mind the inclusion of wordpress.com in your URL, but for reasonable monthly fees, you can use a custom domain and more advanced services on a scale up to $50/month.
This option means you can concentrate your resources on developing and keeping your content up to date. Theme choices are not as wide-ranging as when you self-host using an install from wordpress.org, but you still have hundreds of choices, all of them approved by WordPress and so will work and be safe to use. Most of them nowadays are responsive (adapt to the screen size of the device uses to view the page) as a matter of course. And as time passes if you decide a new theme is needed, you can change by choosing from an ever-growing and updated set of options.
If at any point your organization decides to self-host your site – which gives you total control to adapt the software and also the responsibility for maintaining it through updates and security measures – you can do this by downloading the WordPress software from wordpress.org and installing it on your host. Many hosting sites can do this using an automatic install service.
You can – in this case – transfer the content and settings from your existing site on wordpress.com to your hosted install. This is not something you can easily do with other proprietary sites like Wix, SquareSpace or Weebly, etc.
But your ‘cost/benefit’ assessment for this must include the additional time and cost of security measures, because as the most popular platform on the planet, self-hosted WordPress sites are under constant attack and harassment from spammers to hack them and insert redirecting codes and such. Once you are familiar with managing a WP site it is not difficult to take the steps to maintain security, but it does require some work and constant monitoring.
Hope this helps.