This question is a bit vague and thus difficult to answer - and ironically something I hit my head against this morning. It is impossible to fully do this without the help of your ISP.
The first thing to do is to check your router is allowing all this through - that is the default state for a typical SOHO router though. Depending on your router, you might be able to run tcpdump on it looking at the external interface (eg if its running dd-wrt) and see if packets are leaving your network. If you can't do this, your options are very limited here - a switched on ISP may be able to tell you if they are seeing the packets entering their network - but I would not expect this for your run-of-the-mill home user connection. Alternatively if you can disconnect everything extraneous from your network, you MIGHT be able to get a hint of if traffic is leaving it based on the PPP Interface packet count, or looking at the lights flashing on the modem/router when you attempt to make connections.
Next, if you are able to get onto a server/virtual server which you know is not having connectivity issues, and which has its own "real" IP address or is answering on the server, you can log into that remote box and use tcpdump on the interface to check if it is receiving packets from your machine. If not, then packets are being dropped - if they are being received, then it could be a firewall or issue on the return path.
If you have access to a remote (unfirewalled) server you can do a tcpdump on it while using nmap or netcat to generate traffic to that server and seeing what tcpdump sees.
I don't think this question was vague (though this answer is). We just want to check for the port being blocked or not; tcpdump or netcat is overkill as we're not trying to figure out about packet loss or protocol issues. A simple telnet pass/fail to a server with the port known to be open is a much simpler test. – Michael Altfield – 2018-07-05T17:00:03.760
@maltfield - For a start, an "outgoing port" is a nothing. An outbound connection requires 2 ports - a source and destination port -- both in the same single outbound connection. Your question also does not make reference to the OS, but the answer uses specific OS tools. Your answer - while it may be what you were looking for is also wrong as it ignores UDP. Your question also conflagerates a public network and your network (difference is who controls the routers) and requires use of a third party system which does not eliminate issues on your servers. – davidgo – 2018-07-05T18:37:00.963