If your ISP blocks outgoing connections on port 25 like mine does, then you'll need to have sendmail send through your ISP's SMTP server rather than directly deliver it to the account you're sending to.
If the port isn't blocked, sendmail will do an MX(mail exchange) lookup through DNS, connect to the mail server for the domain you are sending to, and deliver the email to the correct mailbox directly, without going through any other hops(unless the remote machine is a sort of SMTP gateway*).
It can be faster to use your own server if your ISP has a slow, or very busy SMTP server, and it takes a while for your email to get sent.
To use your own server when you aren't at the computer, you will have to use something like SMTP AUTH, which I haven't setup before so I can't help with that. What it does is authenticate you for sending email with your server, and stop it from being used to relay SPAM.
It's important that you setup sendmail to only allow you to send email from it. Otherwise you may find yourself receiving an email from your ISP because people will complain about you sending them SPAM.
If your ISP is any good, they will have more than one SMTP server to accept mail from other ISPs and users, so if one goes down then they will still be able to receive your email. SMTP is also not an "instant" type of thing. The servers are setup to try to send usually for a few days until sending a bounce message if the SMTP server they are trying to send it is temporarily unavailable. The server will usually be fixed by the time that wait period is up.
*what I mean by SMTP gateway is, sometimes very large ISP's will have a few SMTP machines that accept email for the domain, but don't hold the accounts on those machines. They are used usually for spam/virus filtering because that can be a CPU intensive task.
Those machines are setup so that they know where the accounts are and will send the email to those machines after they have been processed.