Generally to setup postfix to relay to Gmail, the first thing you need to do is edit your /etc/postfix/main.cf file to include:

relayhost = [smtp.gmail.com]:587
smtp_sasl_auth_enable = yes
smtp_sasl_password_maps = hash:/etc/postfix/sasl_passwd
smtp_sasl_security_options = noanonymous
smtp_tls_CAfile = /etc/ssl/certs/Equifax_Secure_CA.pem
smtp_use_tls = yes

Then you need to set your gmail password and address in /etc/postfix/sasl_passwd like so:

[smtp.gmail.com]:587    user@gmail.com:PASSWORD

Now make sure it's permissions are correct and update it with:

sudo chmod 400 /etc/postfix/sasl_passwd
sudo postmap /etc/postfix/sasl_passwd

And finally, restart postfix with:

sudo service postfix restart

In your specific case, since you can't telnet to smtp.gmail.com it sounds like you might have your SMTP port blocked. This happens in some specific cases in order to fight against spam. Please open a support ticket, and the team should be able to help you get that fixed: https://cloud.digitalocean.com/support