Is ADO.NET the only data-access method within the .NET Framework?
No, it is not. As mentioned this hour, you can access regular ADO by using the COM Interop objects. Also, you may be familiar with the OLE DB Consumer and Producer templates present in ATL. These can still be used as well. In fact, they have been updated to work with attributes. The downloadable code, which can be found on the book's Web site, contains an ATL Server project that uses the OLE DB Consumer templates with attributes.
When starting a new project, should I immediately rule out ADO in favor of ADO.NET?
If competing technologies are available, you should never immediately rule out one over the other just because it is newer. Look at the pros and cons of each and then make your decision. As an example, if you are writing native code as opposed to managed code, you have to use ADO.
Are OleDbDataAdapter and SqlDataAdapter the only data adapters in the .NET Framework?
They are the only data adapters currently shipping with the .NET Framework. You can, however, download an Open Database Connectivity (ODBC) .NET data adapter from Microsoft's Web site. It's also expected that third-party vendors will create other data adapters.