How can I send emails out from my Ubuntu VPS? I have a domain I want to use (@mydomain.com) for the emailing also. Do I need to set up my own DNS (like bind9 or something)? Or can I use my registrar's DNS (http://name.com) but still yet configure my email server to send emails out using my domain.
This is a pretty vague question and it isn't clear whether you are asking about general best practices, specific software needs, or configuration options. Unless you make the question more specific, expect general answers.
In general you will want to setup DNS hosting somewhere that specifies the MX and SPF records for your domain to use your server. Reverse DNS will also need to work so that other mail servers can validate who you are. You'll need software such as exim, postfix or sendmail to handle the server mail protocols and possibly some sort of authentication method so that you can connect from your computer to the server to send mails so that they come from the server. Of course if you are sending from the server itself that doesn't apply.