Bash - log every input and output and write to file

I am often on assignment and sometimes I forget to document my work. E.g. write down output from different one off tools. Is there a way to automate that via .bashrc? So I open up a new shell, it automatically creates a new log file (like day-month-year-#) and dumps everything in there. Also, it'd be great if I could actually see the output in the window Also: a lot of the output is colorized, is it possible to preserve that?

PS: disk space is a non issue ;)