I found it - an easy way. The command you want is called "script". If you run:
$ script output.txt
it will print the entire session to the text file "ouput.txt" starting for everything after you enter the command. Hitting Ctrl-D will stop script from outputting anymore to the file.
$ man script
for more info on the script command. Here's a quote from the man pages:
Script makes a typescript of everything printed on your terminal. It is
useful for students who need a hardcopy record of an interactive session
as proof of an assignment, as the typescript file can be printed out
later with lpr(1).
Sounds like it's right up your alley.