Yes, whitespace does take up space. Removing or compressing code can sometimes yield up to 95% compression which means it will load that much faster.
There are a lot of tools out there. If you are looking for something very simple just to test it out I would suggest trying an online resource like:
http://refresh-sf.com/
This is a web interface to compress your JavaScript or CSS. This tool uses UglifyJS 2, Clean-CSS and HTML Minifier.
Ideally if you are using some sort of modern tooling process like Grunt or Gulp you can automate the compression as part of your build process.
Google has recently released their version:
https://developers.google.com/speed/docs/insights/MinifyResources
You should minify your HTML, CSS, and JavaScript resources. For minifying HTML, you can use PageSpeed Insights Chrome Extension to generate an optimized version of your HTML code. Run the analysis against your HTML page and browse to the 'Minify HTML' rule. Click on 'See optimized content' to get the optimized HTML code. For minifying CSS, you can try YUI Compressor and cssmin.js. For minifying JavaScript, try the Closure Compiler, JSMin or the YUI Compressor. You can create a build process that uses these tools to minify and rename the development files and save them to a production directory.
Here is a link to Google's "Getting Started with the Closure Compiler Application" that will go over how to compress your files:
https://developers.google.com/closure/compiler/docs/gettingstarted_app