OK it sounds like your ISP does permit outgoing mail. I'm guessing the machine is a shared host and probably NOT on a policy block list. However, this does not mean that mail sent from this machine will actually reach its destination. Shared hosts tend to find themselves on spam blocking lists too because they host a lot of little sites from god-knows-who.
I will also point out that email is not a secure communication channel. This may not be a big deal if you are just delivering contact information to yourself like someone's email and a simple message, but you should never email credit card information or anything sensitive unless it's encrypted. It is possible to set up your server to encrypt mail using this tool. It's a bit complicated, but the explanation in the README file should be pretty informative.
But anyway let me try and get your question answered. PHPMailer supports SMTP mail. I googled around quickly and found an example which I have modified a bit here:
require_once('class.phpmailer.php');
//include("class.smtp.php"); // optional, gets called from within class.phpmailer.php if not already loaded
$mail = new PHPMailer();
$mail->IsSMTP(); // telling the class to use SMTP
$mail->SMTPDebug = 1; // enables SMTP debug information (for testing)
// 1 = errors and messages
// 2 = messages only
// the following settings allow you to use an account on some email system
// to SEND the message. You basically login with the password and send the mail.
// these should work with a gmail account
$mail->SMTPAuth = TRUE; // enable SMTP authentication
$mail->SMTPSecure = "tls"; // sets the prefix to the servier
$mail->Host = "smtp.gmail.com"; // sets GMAIL as the SMTP server
$mail->Port = 587; // set the SMTP port for the GMAIL server
$mail->Username = "yourusername@gmail.com"; // GMAIL username
$mail->Password = "yourpassword"; // GMAIL password
$mail->SetFrom('name@yourdomain.com', 'First Last');
$mail->AddReplyTo("name@yourdomain.com","First Last");
$mail->Subject = "PHPMailer Test Subject via smtp (Gmail), basic";
$mail->AltBody = "To view the message, please use an HTML compatible email viewer!"; // optional, comment out and test
$mail->MsgHTML("<html><body><p>Here is the test HTML message which will be used if your mail client supports html</p></body></html>");
// the recipient
$address = "whoto@otherdomain.com";
$mail->AddAddress($address, "John Doe");
// you can attach stuff if you want
$mail->AddAttachment("images/phpmailer.gif"); // attachment
$mail->AddAttachment("images/phpmailer_mini.gif"); // attachment
if(!$mail->Send()) {
echo "Mailer Error: " . $mail->ErrorInfo;
} else {
echo "Message sent!";
}