I found this url to be very useful: https://social.msdn.microsoft.com/Forums/sqlserver/en-US/2cdcab2e-ea49-4fd5-b2b8-13824ab4619b/help-server-not-listening-on-1433
In particular, my problem was that I did not enable the TCP/IP in Sql Server Configuration Manager->SQL Server Network Configuration->Protocols for SQLEXPRESS.
Once you open it, you have to go to the IP Addresses tab and for me, changing IPAll to TCP port 1433 and deleting the TCP Dynamic Ports value worked.
Follow the other steps to make sure 1433 is listening (Use netstat -an to make sure 0.0.0.0:1433 is LISTENING.), and that you can telnet to the port from the client machine.
Finally, I second the suggestion to remove the \SQLEXPRESS from the connection.
EDIT: I should note I am using SQL Server 2014 Express.