To sign up for an Opportunity you will first need to get to the Opportunity Public Page. You will either be invited to an Opportunity with a sign-up link, or it will appear in your New Signup Opportunities/Search Opportunities tab. By clicking on the preview card for the Opportunity you will be directed to the Opportunity Sign Up Page.
You can always access the Opportunity sign-up page via your Organization homepage. If the Organization Manager has made it so Opportunities are visible, you'll be able to see all upcoming Opportunities on the Organization homepage.