Basically there's a program to accomplish that, called "mail". The subject of the email can be specified with a -s and a list of address with -t. You can write the text on your own with the echo command:
echo "This will go into the body of the mail." | mail -s "Hello world" you@youremail.com
or get it from other files too:
mail -s "Hello world" you@youremailid.com < /home/calvin/application.log
mail doesn't support the sending of attachments, but Mutt does:
echo "Sending an attachment." | mutt -a file.zip -s "attachment" target@email.com
Note that Mutt's much more complete than mail.
You can find better explanation here
PS: thanks to @slhck who pointed out that my previous answer was awful. ;)
Well, the easiest solution would of course be to pipe the output into mail:
vs@lambda:~$ cat test.sh
sleep 3 && echo test | mail -s test your@address
vs@lambda:~$ nohup sh test.sh
nohup: ignoring input and appending output to `nohup.out'
I guess sh test.sh & will do just as fine normally.