By default, most of our customers have email notifications set to go out automatically when tasks/events are created. If you want more granular control over this, it's certainly possible. Below is a screenshot of a task with a "Send Notification Email" checkbox revealed which will allow the ability to choose.
In order to get this checkbox, you'll need to uncheck a setting in Setup, under "Activity Settings". This setting is "Enable User Control over Task Assignment Notifications".
If unchecking this setting doesn't reveal the above email checkbox, you may need to tweak the page layout. Go to the page layout that is in use for either the event or task, then click "Layout Properties" at the top, this will have a couple of options for the email box.