The idea with a try/catch is that if an exception is caught in the try block, none of the remaining code in the try block is valid (i.e. is acceptable to run in this erraneous state). My advice would be to get your hands dirty with try/catch blocks so that you become familiar and comfortable with them, and learn how to use them effectively. If you are worried only about a single line or a few lines of code throwing an exception...
Visual Basic:
Sub Bad()
On Error Goto handler
txtMyTextBox.Text = SomeObject.SomeProperty
Me.Text &= " [Complete]"
Return
Handler:
txtMyTextBox.Text = "(Nothing)"
Resume Next
End Sub
...you can wrap only those lines with a try/catch...
Visual Basic:
Sub Better()
Try
txtMyTextBox.Text = SomeObject.SomeProperty
Catch(Ex As NullReferenceException)
txtMyTextBox.Text = Nothing
End Try
Me.Text &= " [Complete]"
End Sub
The latter only catches an error rising from a null reference. With the first, any kind of expection could be thrown, and if it isn't what you were expecting, something bad could happen.
If you are worried about any one of many lines throwing an exception, all I have to say is that it is very rare that it can be a good idea to plow through a function regardless of what goes wrong. You should generally identify potential problems, reduce chances of them occuring, and handle the errors that do arise in a manner as specific and appropriate as possible. With VB6-style blanket error handling it is very easy to handle the wrong error the wrong way, which is why it is so often discouraged.
But to answer your question: there is no completely analogous method to emulate ResumeNext, except perhaps wrapping each statement in a try/catch block.