How to remove Home link from each new page?

Look for unique CSS for that menu item and hide it per this example and the hints below:

.menu-item-x {display: none;}

To discover what CSS is output by your theme, use a web inspection tool such as Firebug: http://getfirebug.com/ , Chrome Inspect Element: https://developers.google.com/chrome-developer-tools/ or Internet Explorer 9+ F12 Dev. Tools: http://msdn.microsoft.com/en-us/library/dd565627(v=vs.85).aspx#csstool

There are others.

When editing CSS, use a Child Theme
http://codex.wordpress.org/Child_Themes
Custom CSS Plugin, or Theme provided custom CSS option.
Edits to parent themes are lost on theme update.

Learn CSS: http://www.w3schools.com/css/