Need some advice on which way I should go with my website

Okay, I know this will probably create a good bit of controversy, but here goes:

I have a web site which was built a good while ago using the old Microsoft Front Page software. I've been told by the guy who owns my hosting company that in one of Google's many updates, they decided to stop listing sites that were made with this outdated and obsolete software. I've also been told that in one of their latest updates, they are also refusing to list sites that are not "mobile friendly", which means that if a site does not re-adjust it's size to match the size of the screen it's being viewed with, Google will not list it.

I've been using some HTML code to make my site do that, but when I try to bring up my site on my cell phone, it does not re-adjust the size to the screen size. It will re-adjust the size if I bring up my site on a tablet, but that's as far as it goes.

So I guess my main question should be at this point is,

Should I go ahead and re-do my entire web site using the latest version of the Adobe Dreamweaver software ?

Or should I re-do the entire site in a WordPress format ?

If I use the latest version of the Adobe Dreamweaver, I'm going to have to look around and find some kind script or something that will automatically re-adjust the size of the page to the size of the screen it is being viewed with.

Any suggestions here would be greatly appreciated.

If I go with the WordPress idea, I can use a theme (such as the Meeta theme) that has this feature built into it.

Also, if I decide to go with the WordPress idea, will I have to remove all of the HTML pages that were built with the Microsoft Front Page software from my server to avoid any kind of conflicts that could crop up with the various links on any of my pages?

Okay, I think I've rambled on enough for right now and I'm anxiously awaiting any and all responses that will help me decide which way I should go with solving this bit of a dilema.

Thanks, in advance for any and all responses and I really hope that you guys will be able to steer me in the right direction to help solve this problem I'm currently experiencing with my web site.