VisualBasic.NET is a langauge. ASP.NET is a technology for Web Development.
Like you have used vbscript to program ASP, now you would use vb.net to
program asp.net

If you have done programming with VB 6, that would help too..  if you are
finding it difficult to get started with the tutorials available in internet,
probably books like "asp.net is 24 hrs" "asp.net in 7 days"  etc would help..

J2EE Promised: write with one language run on many platforms
.NET Promised: write with many languages run on one platform.

Visual Basic.NET is simply one language that uses the VB syntax and grammar
to develop .NET applications on the Windows platform.

How to start is a broad question. But I suppose the QuickStarts [1],[2] are
as good introduction as any. Note 1.1 and 2.0 are very different in that 2.0
encapsulates a lot of the code we had to write by hand when using 1.1 or
earlier. Starting with 2.0 without some hands-on with 1.1 of study and
application until you really start to master OOP.

<%= Clinton Gallagher
        METROmilwaukee (sm) "A Regional Information Service"
        NET csgallagher AT metromilwaukee.com
        URL http://metromilwaukee.com/
        URL http://clintongallagher.metromilwaukee.com/

[1] http://localhost/quickstart/
[2] http://beta.asp.net/QuickStartv20/aspnet/default.aspx

Get a book on ASP.net
I Recommend starting with asp.net 2.0 as it is quite different than asp.net 1.1

Decide up front if you will be using vb or c# language
Decide if you will develop with Visual Studio or Visual web developer express or just a good text editor

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I would much rather chew on tinfoil than try and code in C#.