Lots of phones have docks that automatically put the phone in "car dock" mode. The OG Droid used a magnet to put it into and out of car dock mode.
The Galaxy Nexus doesn't have any kind of automatic way to start and stop car dock mode, but if your car stereo uses bluetooth, you can use that connection as the trigger. Now I turn on my car and Car Home starts up automatically, and goes away when I turn the car off.
1). Install Car Mode Control. This app will toggle car mode when you run it. Simple and effective. This can be run manually or triggered by another app as I'll show here.
2). Install a Car Dock application. I like the classic Google Car Home, but there are a lot of others out there to pick from.
3). Install AutomateIt. This app allows you to set up a series of rules. They include a Trigger and an Action, and you'll be able to set it up to respond to bluetooth connect/disconnect events.
4). Add a new rule. For Trigger, select "Bluetooth Device Connected" as the trigger, select the bluetooth device that corresponds to your car stero (in my case, it's CZ509, since I have a Clarion CZ509 deck). For Action, select Car Mode Control. It will look something like this:
5). Do the same thing for "Bluetooth Device Disconnected". Action is still Car Mode Control, since that app will toggle car mode.
Now whenever your phone connects to the car (happens automatically for mine), it will switch to car home mode, using whichever car home application you have installed/selected. When you disconnect, it will cancel car home mode automatically.