Save all the terminal output to a file

In the following steps, run xrdb ~/.Xressources after modifying the X-resources file, then open a new terminal to test the changes.

XTerm

Mouse: Hold CtrlLeft-mouse-click. The "Main Options" menu shows up. Select "Print-All Immediately" and release. An XTerm[date] file with the terminal contents has been created in your home directory.

Keyboard shortcut: To bind the action to CtrlShifty, add to your ~/.Xresources file:

XTerm*vt100.Translations: #override
    Ctrl Shift <Key> y: print-immediate()

Setting the path: printFileImmediate sets the prefix of the dumped file. Useful to specify the path of the dumped file. For example, to dump /home/user/dumps/xt[date] files,

XTerm*printFileImmediate: /home/user/dumps/xt

Urxvt

Keyboard shortcut: Include in ~/.Xresources:

URxvt.print-pipe: cat > $HOME/scrollback

Binds to Ctrl+Print or Shift+Print.

Other terminals

Not all of them can do it; check their manual page or menu options.

Beyond script, a terminal-agnostic solution is to use a multiplexer.

In Tmux, it's a matter of running its internal command

capture-pane -S - ; save-buffer scrollback-file

This in .tmux.conf would bind it to Ctrl+bCtrl+s:

bind C-s capture-pane -S - ; save-buffer $HOME/scrollback

I too faced the same problem and after some search came up with this solution:

Add to your .bash_aliases this:

# Execute "script" command just once
smart_script(){
    # if there's no SCRIPT_LOG_FILE exported yet
    if [ -z "$SCRIPT_LOG_FILE" ]; then
        # make folder paths
        logdirparent=~/Terminal_typescripts
        logdirraw=raw/$(date +%F)
        logdir=$logdirparent/$logdirraw
        logfile=$logdir/$(date +%F_%T).$$.rawlog

        # if no folder exist - make one
        if [ ! -d $logdir ]; then
            mkdir -p $logdir
        fi

        export SCRIPT_LOG_FILE=$logfile
        export SCRIPT_LOG_PARENT_FOLDER=$logdirparent

        # quiet output if no args are passed
        if [ ! -z "$1" ]; then
            script -f $logfile
        else
            script -f -q $logfile
        fi

        exit
    fi
}

# Start logging into new file
alias startnewlog='unset SCRIPT_LOG_FILE && smart_script -v'

# Manually saves current log file: $ savelog logname
savelog(){
    # make folder path
    manualdir=$SCRIPT_LOG_PARENT_FOLDER/manual
    # if no folder exists - make one
    if [ ! -d $manualdir ]; then
        mkdir -p $manualdir
    fi
    # make log name
    logname=${SCRIPT_LOG_FILE##*/}
    logname=${logname%.*}
    # add user logname if passed as argument
    if [ ! -z $1 ]; then
        logname=$logname'_'$1
    fi
    # make filepaths
    txtfile=$manualdir/$logname'.txt'
    rawfile=$manualdir/$logname'.rawlog'
    # make .rawlog readable and save it to .txt file
    cat $SCRIPT_LOG_FILE | perl -pe 's/e([^[]]|[.*?[a-zA-Z]|].*?a)//g' | col -b > $txtfile
    # copy corresponding .rawfile
    cp $SCRIPT_LOG_FILE $rawfile
    printf 'Saved logs:n    '$txtfile'n    '$rawfile'n'
}

And to the end of your .bashrc file add this:

smart_script

After you've done this, "script" command will be executed once in every terminal session, logging everything to ~/Terminal_typescripts/raw.

If you want, you can save current session log after the fact (in the end of the session) by typing savelog or savelog logname – this will copy current raw log to ~/Terminal_typescripts/manual and also create readable .txt log in this folder.  (If you forget to do so, raw log files will still be in their folder; you'll just have to find them.)  Also you may start recording to a new log file by typing startnewlog.

There will be a lot of junk log files, but you can clean old ones from time to time, so it's not a big problem.

(Based on https://answers.launchpad.net/ubuntu/+source/gnome-terminal/+question/7131 , https://askubuntu.com/a/493326/473790 )

You can use script. It will basically save everything printed on the terminal in that script session.

From man script:

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).

You can start a script session by just typing script in the terminal, all the subsequent commands and their outputs will all be saved in a file named typescript in the current directory. You can save the result to a different file too by just starting script like:

script output.txt

To logout of the script session (stop saving the contents), just type exit.

Here is an example:

$ script output.txt
Script started, file is output.txt

$ ls
output.txt  testfile.txt  foo.txt

$ exit
exit
Script done, file is output.txt

Now if I read the file:

$ cat output.txt

Script started on Mon 20 Apr 2015 08:00:14 AM BDT
$ ls
output.txt  testfile.txt  foo.txt
$ exit
exit

Script done on Mon 20 Apr 2015 08:00:21 AM BDT

script also has many options e.g. running quietly -q (--quiet) without showing/saving program messages, it can also run a specific command -c (--command) rather than a session, it also has many other options. Check man script to get more ideas.