I'm trying to send email from my home mail server. It works fine for incoming email, but any time I try to send I get rejection messages from some recipients. What's going on?
If you were having problems with sending to all recipients it would suggest that your ISP was blocking the email port (25/TCP). That you can send to some but not all suggests your problem is simple - you're trying to send directly.
Few providers these days will accept email directly from a dynamic IP address. To send email you either have to be on a static IP address (and some providers static IP ranges are just subsets of their dynamic allocation and so are still treated as if they're dynamic) or relay your email through a mail server. Your options there are to use your ISP's mail server or a service.