Configure Zimbra to use SendGrid as Smarthost

# Switch to user Zimbra
su zimbra

# Add credentials to relay_password file:
vi /opt/zimbra/conf/relay_password
# Add: yourSendGridUsername:yourSendGridPassword

# Apply configuration to Zimbra
postmap /opt/zimbra/conf/relay_password
postmap -q /opt/zimbra/conf/relay_password
zmprov ms `zmhostname` zimbraMtaSmtpSaslPasswordMaps lmdb:/opt/zimbra/conf/relay_password
zmprov ms `zmhostname` zimbraMtaSmtpSaslAuthEnable yes
zmprov ms `zmhostname` zimbraMtaSmtpCnameOverridesServername no
zmprov ms `zmhostname` zimbraMtaSmtpTlsSecurityLevel may
zmprov ms `zmhostname` zimbraMtaSmtpSaslSecurityOptions noanonymous
zmprov ms `zmhostname` zimbraMtaRelayHost

Now wait for a couple of minutes and start testing it by sending out some test mails.

Follow the log file to see if it works or not:
tail -f /var/log/zimbra.log