What do I need to do to be able to access my Plex Media Server on my laptop when I'm away from home
The best way to establish a remote connection between Plex Media Center on your laptop and your Plex Media Server at home is by using the myPlex service. myPlex greatly simplifies the process of remotely connecting your iOS device to your media server. Just go to my.plexapp.com and if you already have a forum account you already have a myPlex account. If not just setup an account on the myPlex website.
In Plex Media Center go to Preferences > System > myPlex
Enter the email address and password you used to setup your myPlex account and then select select "Sign In". Plex Media Center will connect to your Media Server at home and will show all of the library sections you created on the main menu. You do not have to have Plex Media Server running on your laptop for this to work.