I am trying to connect to the oracle database which contains all the government environment related spatial data from my Arc GIS desktop so that I can always get the latest/ updated layers.
You may connect to and work within enterprise geodatabases using ArcDesktop Standard or Advanced versions.
In order for ArcDesktop to connect to an Oracle geodatabase, each machine with the connection ArcDesktop application must have the Oracle database client installed. The client is basically a translator between the Oracle and Esri softwares, allowing them to speak to each other.
ArcDesktop is a 32 bit client, so you will need the 32 bit Oracle database client installed on there. You may choose the instant or full client (your dba should have a preference for you). If you also need to use ArcGIS Pro or connect to ArcGIS Server (both are 64 bit applications) then you will also need the 64 bit Oracle database client installed on those same machines. The 64 bit client should be listed first in the PATH environment variables.