Use the string concatenation operator:
Dim str As String = New String("") & "some other string"
Strings in .NET are immutable and thus there exist no concept of appending strings. All string modifications causes a new string to be created and returned.
This obviously cause a terrible performance. In common everyday code this isn't an issue, but if you're doing intensive string operations in which time is of the essence then you will benefit from looking into the StringBuilder class. It allow you to queue appends. Once you're done appending you can ask it to actually perform all the queued operations.
See "How to: Concatenate Multiple Strings" for more information on both methods.