I would like to change which icons appear on the dock when the computer is first started. I'm hoping to not have to drag the icon off the dock every time I restart the computer.
That's not normal behavior. Your dock should keep your saved preferences during a restart
Try removing the dock's preferences then then customize it back to how you like it then try and restart and see if it is fixed
To remove the dock's preference file, switch to Finder, then select "go to folder" in the "Go" menu. Then type in the box
Code:
/Users/yourusername/Library/Preferences/com.apple.dock.plist
replacing "yourusername" with the name of your user respectively. Then delete that file and log out or restart.