What is the difference between installing the full Oracle Client and an Oracle Odac install? Which will I need to do .NET development on my dev workstation and which will I need on the web application server that will connect with the Oracle database on another server?
ODAC includes Oracle Data Provider for .NET, Oracle Developer Tools for Visual Studio (ODT), Oracle Providers for ASP.NET, .NET stored procedure support, as well as additional Oracle data access software for Windows.
-Oracle's site.
The full Oracle client includes a lot of extra software, for example SQLPlus, SQL Developer etc.
Anyways, that's mostly irrelevant. The standard way of obtaining database drivers in .NET is by using NuGet, and Oracle has an official managed driver with no outside dependencies that's the simplest way to get started. You add that package via NuGet to your application, and the necessary DLL's will be included with your application when you publish it without needing to specially configure the server.