You can try Boostrap framework, it allows you to create fast responsive templates without knowing too much about programming.
Also, there are many existed templates and you can use it for your websites.
If you have a good budget, go with Wordpress + Themeforest, you can change different responsive template in seconds with some clicks.
If you're interested in learning responsive web development, it's basically just like normal web design but you make it in a way that fits most screen sizes. Mostly by using CSS media queries to change the layout and positioning of elements on your web page. As mentioned by christax, you can use something like Bootstrap to easily create responsive designs.
If you're not a developer and just need a responsive template for your website, you can find a lot of them online. Try searching on Themeforest. There might also be free ones when you search on Google.
Its something you will need to learn even when editing templates with bootstrap framework things sometimes do need adjustments.
Media queries are the "go to" here is a easy tip -
1)Open your page in your browser
2) go into developer mode
3)start downsizing the window until the layout starts messing up ( note the screenwidth)
4)adjust the css rules IN the browser dev tools until things look good
5) go set a new media query at that width and paste the rules ( you created in the browser) in your css file under the new media query
6)repeat until you get down to 320 px width.
Be sure to check for browser compatibility too.
Also checkout Fluid grid layouts im not a big fan of them but maybe it will work for you.