Although it is possible to make responsive HTML/CSS with nothing more than your bare hands and a decent understanding of CSS media queries and a lot patience I wouldn't recommend that approach to anyone. Choose a responsive CSS framework. It's grid will do the tedious work of giving you responsive layout classes. My personal favorite is Foundation Frameworkhttp://foundation.zurb.com/ because it comes with many responsive components aside from a grid. Skeletonhttp://www.getskeleton.com/ is more lightweight, it's just a grid, a responsive form of the 960 grid I believe. Others use Twitter Bootstrap. There are probably 5-10 more that you could checkout but those come to mind for me. Once you find one you like and know it, you might want to stick with it. I use Foundation whenever, wherever possible.

Now just using a Framework alone does not mean you will have a good responsive site. Frameworks are not magic, they just give you a responsive layout. You must still understand responsive design and apply responsive design principles, and test heavily in all screen sizes. Generally the approach of "mobile first" is exactly that, layout your design in 1 column or at most 2 columns and make sure it looks great at about 320px wide. Watch for any images that don't scale to that size. Watch for any elements that by design are too wide to scale nicely and look garbled at 320px.

Foundation provides a grid where options look like "large-4" and "small-12". So if I make the following divs it will use 4 columns out of 12 on a large screen, and all 12 on a small screen. So basically on larger screens in the example below it will be 3 columns beside each other. But on mobile each of those divs will be full width of the screen because of the "small-12" class.



Generally we never want more than 2 columns on a small mobile device unless we're layout out some small buttons but if you split 320px across 3 columns it's only 110 or pixels each. Very small. So normally I would mostly use "small-12" to make all my divs collapse into 1 column. But I could use the class "small-6" to say use half the available width.

Hope you find this helpful, also consider checking out YouTube for responsive design videos there are quite a few that show specific site examples and go quite deep into the topic.