mysqldump & gzip commands to properly create a compressed file of a MySQL database using crontab

First the mysqldump command is executed and the output generated is redirected using the pipe. The pipe is sending the standard output into the gzip command as standard input. Following the filename.gz, is the output redirection operator (>) which is going to continue redirecting the data until the last filename, which is where the data will be saved.

For example, this command will dump the database and run it through gzip and the data will finally land in three.gz

mysqldump -u user -pupasswd my-database | gzip > one.gz > two.gz > three.gz

$> ls -l
-rw-r--r--  1 uname  grp     0 Mar  9 00:37 one.gz
-rw-r--r--  1 uname  grp  1246 Mar  9 00:37 three.gz
-rw-r--r--  1 uname  grp     0 Mar  9 00:37 two.gz

My original answer is an example of redirecting the database dump to many compressed files (without double compressing). (Since I scanned the question and seriously missed - sorry about that)

This is an example of recompressing files:

mysqldump -u user -pupasswd my-database | gzip -c > one.gz; gzip -c one.gz > two.gz; gzip -c two.gz > three.gz

$> ls -l
-rw-r--r--  1 uname  grp  1246 Mar  9 00:44 one.gz
-rw-r--r--  1 uname  grp  1306 Mar  9 00:44 three.gz
-rw-r--r--  1 uname  grp  1276 Mar  9 00:44 two.gz

This is a good resource explaining I/O redirection: http://www.codecoffee.com/tipsforlinux/articles2/042.html

if you need to add a date-time to your backup file name (Centos7) use the following:

/usr/bin/mysqldump -u USER -pPASSWD DBNAME | gzip > ~/backups/db.$(date +%F.%H%M%S).sql.gz

this will create the file: db.2017-11-17.231537.sql.gz

You can use the tee command to redirect output:

/usr/bin/mysqldump -u user -pupasswd my-database | \
tee >(gzip -9 -c > /home/user/backup/mydatabase-backup-`date +\%m\%d_\%Y`.sql.gz)  | \
gzip> /home/user/backup2/mydatabase-backup-`date +\%m\%d_\%Y`.sql.gz 2>&1

see documentation here