Setting up an Office365 account is a very straightforward process. Very quickly you can be up and running with Online Exchange and Lync. If you wish to use your own domain name instead of an onmicrosoft.com name then you will need to add that domain to Office365 and modify half a dozen DNS records in your domain (full instructions are included on the Office365 site).
However, the default configuration will not allow you to use the Lync client to communicate with External contacts. The error message when you try is none too helpful. To enable this you will need to go the Office365 Lync Control Panel and under Organization select the External Communications tab. Here you can change the setting to allow external access.