A starndard block for the front page of your public site is "Feature Events". This block shows the next 3 upcoming events. Once an event has passed, it will no longer appear here (pro tip: it's a good idea to write a new/blog item about past events!).

Add or edit an event

• to add an event, go to settings & admin > add content > event
• to edit an event, go to the event and click the edit tab

Edit the event information

Make sure it has a title, event type, date and location. You can fill out as much or as little information as you want, but a title, date/time and event type are required.
It's important, however, for events that should be featured on your public site, checkmark BOTH "Public Event" & "Featured Event". If it's only marked as "public" it will appear on your public site on the events page, but not in the "featured events" block.

Edit the public event information

When you see the "Public Events Fields" area, click on the title to open the options. Create and upload an event image (templates are in here) -  make sure that the image matches the template dimensions. Add public notes - this is what the visitor to your site will see - keep in mind that the front page shows only a snippet of this so the first sentence should be your most compelling.

 

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Go to your front page. Is your event showing? Should it be?

Remember, events only show up in the featured events block if they're marked as both public and featured.

If your event is not showing, simply flush the site cache. To do this, search for "cache" in the admin menu search OR go to settings & admin > advanced > flush caches

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Have a look at this related article that discusses ideas for featured events.