Port 465: IANA has reassigned a new service to this port, and it should no longer be used for SMTP communications.
However, because it was once recognized by IANA as valid, there may be legacy systems that are only capable of using this connection method. Typically, you will use this port only if your application demands it. A quick Google search, and you'll find many consumer ISP articles that suggest port 465 as the recommended setup. Hopefully this ends soon! It is not RFC compliant.
Port 587: This is the default mail submission port. When a mail client or server is submitting an email to be routed by a proper mail server, it should always use this port.
Everyone should consider using this port as default, unless you're explicitly blocked by your upstream network or hosting provider. This port, coupled with TLS encryption, will ensure that email is submitted securely and following the guidelines set out by the IETF.
Port 25: This port continues to be used primarily for SMTP relaying. SMTP relaying is the transmittal of email from email server to email server.
In most cases, modern SMTP clients (Outlook, Mail, Thunderbird, etc) shouldn't use this port. It is traditionally blocked, by residential ISPs and Cloud Hosting Providers, to curb the amount of spam that is relayed from compromised computers or servers. Unless you're specifically managing a mail server, you should have no traffic traversing this port on your computer or server.