Current session:
You can do so by running the following command in the terminal:
history -a ~/current_session.txt
Current session's history will be saved to a file named current_session.txt in your home directory.
Certain inputs:
You can also search all history for a certain input and save the output to a file. For example to save all lines that have install in them, please run the following command in the terminal:
history | grep install > ~/search_results.txt
Search results will be saved to a file named search_results.txt in your home directory.
Change install to whatever you want to search for.
To search for multiple inputs put them between two quotation marks
"" , separate them with a pipe | and put -E before them like so:
history | grep -E "install|update|upgrade" > ~/search_results.txt
Best of luck