I couldn't get Dennis' very simple one-liner to work, so here's a far more convoluted method. I'd try his first.
As mentioned, you can use exec to redirect standard error & standard out for the entire script. Like so:
exec > $LOGFILE 2>&1
This will output all stderr and stdout to $LOGFILE.
Now, since you want to have this displayed to the console as well as a logfile, you're also going to have to use a named pipe for exec to write to, and tee to read from.
(Dennis' one-liner technically does this as well, although obviously in a different way) The pipe itself is created with
mkfifo $PIPEFILE. Then do the following.
# Start tee writing to a logfile, but pulling its input from our named pipe.
tee $LOGFILE < $PIPEFILE &
# capture tee's process ID for the wait command.
# redirect the rest of the stderr and stdout to our named pipe.
exec > $PIPEFILE 2>&1
echo "Make your commands here"
echo "All their standard out will get teed."
echo "So will their standard error" >&2
# close the stderr and stdout file descriptors.
exec 1>&- 2>&-
# Wait for tee to finish since now that other end of the pipe has closed.
If you want to be thorough, you can create and destroy the named pipe file at the start and end of your script.
For the record, I gleaned most of this from a random guy's very informative blog post: (Archived version)