I eventually managed to solve this issue. By default, after installation, the TCP/IP connection had been disabled. I needed to open up 'SQL Server Configuration Manager', go to the network configuration section and enable TCP/IP.
Under the 'IP Addresses' menu on TCP/IP I also needed to make sure the IP4 option was active, and the IPAll port matched what I expected the server to be listening under.
hot2use's answer had a lot of potentially useful options for figuring out the issue, but in the end none of them were able to diagnose my exact problem. All results returned from querying the server itself said everything was working correctly, it wasn't until I looked at this tool that I found some settings were not as I expected.
