I agree with everything Bobkin said, and also welcome you to VMware. I hope you stay!
To start with VMware, you first interact with the DCUI (Direct Console User Interface). This gives you the ability to change the TCP/IP stack on the host. You're probably there already but just checking. Then from a remote computer type the IP address into the browser that you see on the ESXi's DCUI. This allows you to download the vSphere Windows Client or to interact with the Web Client. Even if you are on something as strict as a sensitive government network, this would still work because you don't need web connectivity: the ESXi has these downloads built into it.
Installing vCenter is a whole different subject. You would be using the VCSA (vCenter Server Appliance) which is built on SUSE Enterprise Linux. The ways to install this can be profoundly different depending on which version you're using. But unlike the Windows vCenter installation which require putting the installation CD into the hypervisor itself (or attaching it through the datastore), you just burn the files onto a CD and then put it into your workstation. You will need to download them from your VMware account which means that they will not be made available from the hypervisor like the vSphere client.
However, you don't need vCenter to activate shell access, you just need to go into the configuration for your host and change the settings under your host's security profile like Bobkin said.