Step 1: Put the BeagleBone on my home network with the CAT5 cable from the router. It gets a DHCP address on the network automatically from the router.
Step 2: Go to root user.
sudo su
Step 3: Update apt-get, which synchronizes what packages my distro has and doesn't have and establishes locations to download from.
apt-get update
Step 4: Download and install Tkinter. Despite what I read online about other users' issues with this step, it seemed to work fine for me.
apt-get install python-tk
Some related links:
The basic instructions, which include the above steps http://tkinter.unpythonic.net/wiki/How_to_install_Tkinter
About how to use apt-cache search which while not specifically needed to do this installation is generally helpful. Also, aptitude is mentioned: http://askubuntu.com/questions/160897/how-do-i-search-for-available-packages-from-the-command-line
More about apt-cache search, and also some more about how to use aptitude: http://www.cyberciti.biz/faq/searching-for-packages-in-debian-ubuntu-aptitude/