How do I edit the format of things published to the front page? How do I change the way info is published to the front page?

For example, it gives me a title, 1 paragraph of body, date, tags and a read more button.  I need to set it so that it also gives me a particular image field from a custom content type on top of the body content.  

Where do I make this correction?  I don't see it anywhere in views and I don't want to redesign my front-page around a different block / new view.
It also possible with Views, if you set the Style (listed in the Basic Settings section) to Fields. You can then choose whatever Fields you like to be displayed.

Like you already mention yourself, it is possible with Display Suite. If you choose to work with Display Suite, you can select a Display Suite build mode in the Style option of a View.