Hello PapaWhiskey, and welcome to Seven Forums.
Since you have your taskbar dragged up with two levels high instead of the single level, you will see the two icons (Explorer and Homegroup) displaying from the "Desktop" toolbar like that. If you only want to see the "explorer" icon, then you would need to shrink (drag) the taskbar back down to one level.
As for the chevron arrows menu, you can get rid of it by dragging the dotted separator line to the left until all the chevron arrow goes away. If you keep the chevron arrows, there's nothing that you can do to remove the menu names and icons when you click on the arrows. This is just part of the "Desktop" toolbar.
When finished, lock the taskbar
You might find this to be helpful as well. Toolbars - How to Use in Windows 7 and Vista
Hope this helps some,