Put simply
If you can learn to develop Windows forms in VB.NET, and get accustomed to data handling, using datasets, tables (ADO.Net), then you will find it very easy to jump to creating a data based web page in ASP.Net. Some of the Windows and Web based controls are very similar in operation. Also .Net is very much about disconnected record sets, which lends itself very much towards Web based development also.
In ASP.Net, all the VB.Net code you write is executed on the Server, and any Client scripting is still done using Javascript or VB Script. You also need some knowledge of HTML, to be able to beautify your pages.
In my opinion, you have enough to learn swapping to VB.NET from VB6. Get a good grounding here, and dont try to run before you can walk.
Sweep
...if it works dont mess with it