Get a list of network connections my machine is making

You can't get ended connections unless you were logging them. To get the current connections:

# sudo netstat -tupn

This will show the current opened source and destination ports with IPs.

To get more details about the contents of the connections, install tcpdump if it's not installed and run:

# sudo tcpdump -X -i eth0

And you can run that in the background and write the output to a file.

If you are loooking for a list of ports you can use:

netstat -tupn | grep -E ":(22|80|8000) .*"

netstat -tupn | grep -E ":(22|80|8000) .*ESTABLISHED"

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