Build a Responsive Website

I am looking to modernize and completely update an existing website that I am responsible for. To get an understanding of the current website...go to deepcutorchids.com. Be warned its as ugly and unfriendly to use as it gets...I just inherited the management of it. This is for a non-profit orchid club in NJ. As you will see the site is ugly, date, and not mobile or tablet friendly. As part of this process I also plan to add a .org URL which I plan to register as part of this process (and keep /redirect from the .com URL ) . I am not opposed to stay hosted with Google sites..but am not married to it...I am thinking of registering the new URL and hosting with godaddy, or something like that, and in the future, deploy URL branded email to the club members. I would like the site to become more like this... and [login to view URL] or this [login to view URL] but with its own unique personality. This is early stage, and I'd be ok to iterate over time if it makes sense, but the ultimate end state should be something I can maintain. I am by no means a developer but have been in IT for 15+ years and have designed and managed websites for other non profits in the past....using Dreamweaver...but long before smart phones and mobile browsing became prevalent. I would like to agree on a ballpark estimate and see some examples of works completed.