Need to make RESPONSIVE WEBSITE NEEDED- Adjustments to existing site

I presently have a website template with a company called Bludomain. I need to host with another company as well as make changes to this template. It isn't responsive at all and when viewed on various screens the website looks different depending on the device. For instance on one computer it will look perfect, then another the text falls off the page or a scroll bar can't be seen etc.

I want my website to be formatted to fit all types of screens as is my original business website (which I have NO problems with).

I am uncertain if this is possible and what it would entail to do so. I preferably would like to have a responsive template for what I currently display on my site without making changes on the visual side of things. I had my individual programmer add in an uploading form however other than the entire template is standard for Bludomain Possibly Wordpress would be a good template option for me

Desired Skills are MySQL Administration, HTML, PHP