Save all the terminal output to a file

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.

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 )