You want an SMTP relay. You can use a free public SMTP relay, or make one yourself with something like Postfix. If you make one yourself, don’t keep it open, or spammers will abuse it. If you’re sending mail properly, you’ll need to add it to your SPF entry in your DNS. If you’re just sending mail for yourself, you should be able to set up a filter in your email provider so that it doesn’t get rejected.
You also need to know if outbound traffic is blocked on port 25 by your ISP. Most ISPs block it to prevent spam. If yours is blocked, you can request that they unblock it, but they might not do it. If they don’t, you’re kind of SOL. At that point, you need an account somewhere to send mail through. Then you’d be connecting on port 587 to submit mail.