Allow users to filter through categories of custom post type (should use php & jQuery - no plugins)

Hi. We have a page called "our work". This page shows all kinds of our projects. These projects were built by registering a custom post type in our [login to view URL] file - the custom post type is named "project". Each project is given a category. The category functionality is built by registering a custom taxonomy for the custom post type "project". It's rather straight forward. All I am trying to do is display these categories on my work page, AND allow users to filter through these categories. So if one project category is selected - the other projects categories should disappear until the user clicks on that category. Make sense? See screenshots for details. This shouldn't take long for someone who knows what they are doing. I don't need you to style the category menu you are building... it can be an ugly unordered list for all I care... I can do the CSS and styling. I just need you to build the functionality into the page. I'll attach screenshots of some of the existing important code and a screenshot of what the page is intended to look like.