I'm new to Visual Basic 6.0. Can I start with Visual Basic .Net or
should I begin with Visual Basic 6.0. instead?
They are nearly different languages, so you should do one or the other, but not both until you have a good grasp of one and can recognize the differences.
I'd go for VC .NET and ignore VB 6 for a couple years. If you are planning to get a job as a VB programmer then you will need both because there is a lot of VB6 code that you will probably have to maintain.