The default Cygwin terminal uses the Windows terminal. If you have ever used the Windows terminal, you will know that it not support some of the basic functionality found in other terminals like Xterm. To get a more featureful terminal without installing X, install the Rxvt package. Then replace the contents of your cygwin.bat file with the following batch script. This example runs a bash login shell when the terminal starts. Note that the login (-l) flag is important because it tells bash to source /etc/profile which in turn will source the user's .bash_profile file.
rxvt -bg black -fg white -fn "Lucida Console"-11 -geometry 110x75 -e /bin/bash -l -i