How do I get terminal access while leaving my Pi at home?
The first thing to know is your RPI2's username, IP address, and hostname.
By default,
the username is pi and the hostname is raspberrypi. You can
look up your IP address with the command ifconfig.
The default Raspbian distribution for the RPI2 comes with an SSH server pre-enabled and running. If you are connecting to your RPI2 that is sitting on the same network as you are, then depending on how your network is configured, you should be able to just ssh to pi@(hostname), or failing that, ssh to pi@(IP address).
If you have an office in the CSE department somewhere, the campus network makes this easy. First, you will have to change your hostname (because there are a lot of students with the default raspberrypi, and your device will collide with theirs). Second, just plug your RPI2 into the network and it will automatically recieve the address (hostname).seas.wustl.edu. To connect, just ssh to pi@(hostname).seas.wustl.edu. (NOTE: You can only connect to this address if you are inside the WashU firewall. To connect from off-campus, you will first need to ssh inside one of WashU's internet-facing servers, and then you can ssh to your RPI2 from there. For example, when I connect from off campus, I first ssh into ssh.seas.wustl.edu, and then hop over to (hostname).seas.wustl.edu.
If you would like to set up your RPI2 at home for remote access, you'll need access to your internet router. The difficult thing about this is that your RPI2 is given one IP address by your home router, while your home is given a different IP address by your internet service provider. In order to make this work, we need to configure your router so that ssh requests are forwarded to your RPI2. By default, ssh communications are sent over port 22, so we need to forward port 22 to your RPI2. Go into your router's advanced settings and look for port forwarding (not port triggering, which is different). You will need to add a custom rule that forwards port 22 to the internal IP address of your RPI2. Then, use your router's configuration screen to look up what external IP address it has been assigned by your internet service provider. In order to connect to your RPI2 while away from home, you should be able to ssh to pi@(external IP address).
Note that if you are connecting to home, there is no guarantee that your internet service provider will assign you the same IP address for long periods of time. If this is the case, your external IP address may change frequently, making the above approach difficult. There are two possible solutions. First you can try calling your internet company and asking for a static IP address. If they say no, or if you would prefer a more robust method, you can see if your router supports a Dynamic DNS service. If so, there are many organizations that will give you a path (like david.noip.com) that will always map to your home, no matter how often your external IP address changes. There are organizations that will do this for free, and some router manufacturers (like Netgear) offer this as a free service if you use one of their routers.
How do I get a remote graphical interface while leaving my Pi at home?
First, go through the steps for setting up a terminal interface at home (above).
A free, open-source solution for getting remote graphical access to your Pi is via VNC (Virtual Network Computing). You will need to install a VNC server on your PI, and get a VNC client (viewer) in order to connect to your server. An excellent open-source implementation of VNC is called TightVNC, which provides both a server and a client.
The folks at raspberrypi.org have an excellent walk-through of how to install, configure, and automate the TightVNC server. This only takes a few minutes to install and configure. You can access it here. Note that if you plan on sitting down and using your Pi through a regular keyboard and monitor at home, in addition to connecting remotely, you probably don't want to disable the existing display :0
Once you have that complete, you need to install the TightVNC viewer on the machine that you'll use to connect to your Pi. The link above has instructions if you're connecting from a Linux machine. If you're connecting from Windows, go to tightvnc.com and download their viewer from there. If you're connecting from Mac OS, or from any machine that you can't install software on, you can go to tightvnc.com and download their self-contained Java viewer (which of course requires Java to be installed).
Finally, if you're connecting from outside of your house, you'll need to forward ports 5900-5902 on your home router to your Pi. (See previous answer for details.)
When connecting with a viewer, you will connect to (external IP address)::(port number). The port number is determined (by default) to be 5900 plus the number of your display. So, if your VNC server is running display :0, then your port number is 5900. If your VNC server is running display :1, then your port number is 5901, etc.