To find an event's page for use on other sites, start by navigating to your pages URL (found at Main Menu->Website Integration->Pages) and navigating to the Events tab. From there you can either use the URL for the events page (taken from the browser address bar) or navigate to a given event.
Note that the calendar will start with the current month, so if you don't have an event this month you may need to navigate to the month you have an event within.
Once you navigate to an event, simply copy the URL from the address bar. Even when you are logged in as an Admin, the URL will be the same for you and for contacts.
To appear online an event must be configured and set to one of the Publish option. Publish Now is the default.