Michelis,
Try sending to another Gmail account, first. That will tell you if you're successfully getting into the Gmail SMTP server, or not. If you receive the mail at your alternate Gmail account, then you've made it this far. If not, you're still missing something in the connect/send bit on your side.
If your test is received at another Gmail account, but not at a Hotmail, Outlook or Yahoo account, then you might only be missing some security settings in the connect bit on your side. This is the kind of troubleshooting I do with my clients who have the same problem.
Flint