I don't know how much of your problem is dependent on Sketch, does it output to html? I do know the basics of responsive web design though.
The Creative Style Sheet (CSS) is linked from your HTML web page. The style sheet contains Media Queries which check which screen size the user has and delivers styles for that size.
They look like this:
@media only screen and (min-width: 600px) {
/* For tablets: /
{Styles defining the columns and spacing and everything you want for tablet size}
}
@media only screen and (min-width: 768px) {
/ For desktop: */
{Styles defining the columns and spacing and everything you want for desktop size}
}
There are several more sizes: phone, tablet horizontal and vertical, laptop and desktop, big screen etc. They are all defined by pixel dimensions. When a certain device opens a web page it gets those styles that work on their screen size.
That's the responsiveness!
Most basically you can design layouts for each screen size and put each in the media query for that screen size. They can be completely different styles, proportions, colors, layout etc.
Usually people use the same format laid out differently on different screen. For example where they have 12 columns on a desktop they'll have 3 columns on a tablet and 1 columns on a phone. The content in the remaining columns is shifted below the top row so you scroll down to see them.
Programming all the responsiveness is pretty complicated for a beginner but there are pre-defined media queries in web platforms like bootstrap, which is free.
If you have the bootstrap CSS linked up to your web page you can use their pre-defined code to set your columns on different screen sizes. They use a max 12 column scheme, so after you start a row you can start a column with class "col-md-12" which means there will be 12 columns on medium screens and it will auto shift them down on smaller screens. A column can have multiple classes if you want get specific you can classify one col-lg-12 col-md-3 hidden-sm which means 12 columns on a large screen, 6 columns on a medium screen and hide it on small screens.
The gutter, or spacing between columns can also be defined the same way, within a media query for large screens make a rule something like padding-right: 20px. and in the media query for small screens make the padding less.
I believe Bootstrap gives the same spacing between columns on large and small screen sizes. It doesn't look funny because 10 pixels is a lot smaller on a phone than on a desktop.
I hope this helps.