I realize that VB.NET and ASP.NET are the successors of their seemingly obsolete classic counterparts, but I have noticed that some job descriptions still list VB6 and classic ASP as requirements. This may be because they want people who can convert code from classic to .NET, but I'm not really sure. Would classic VB and ASP be helpful in maintaining a long-term career, or should I just skip them entirely and learn .NET? Are the differences between the old and new enough to even make a difference as to which I learn first? If it helps, I'll be looking to start an internship in about a year and a half, but I'm not going to be looking for an actual job for another year or two after that.
If a job requires VB6 and classic ASP I'd run. Unless you're about to retire, you should stay less than 10 years back in development and learn new technologies.
Learn .NET, preferably C#.NET (easy if you know Javascript). VB and classic ASP will be easy to figure out after that.
Whether you should learn VB6 or VB.NET depends on your goal. VB6 is easier to learn so if you want to learn any programming language, it will be easier to learn VB 6. Many people also work for companies that have a large embedded base of VB 5 and VB 6 code, and they need people to support that code indefinitely. If you are learning to program so you can help support this code, you need to learn VB 6.
Many of the concepts in VB .NET are different from VB 6, however. VB .NET is more object-oriented and follows a model closer to the one used by C++ and C# (no coincidence). Learning VB 6 first won't help you too much with those concepts and you may be better off learning them first so you are a "native" object-oriented programmer. If you later plan to learn other languages such as C++ or C#, you may want to start with VB .NET.
The 2 are greatly different. Unless you are forced to use classic ASP, you should go for ASP .Net. If you are going the .Net route, it would probably be worthwhile to go with C# as the base language as well.
Learn .NET. They're migrating to it for a reason: it's the "latest and greatest" and has more advantages than its predecessors. Think of it this way: 10 years from now when we're all migrating .NET to XYZ (hypothetically speaking) the VB6/ASP knowledge won't be helpful.
Why are you willing to pigeon-hole yourself into that segment of the market to begin with? I've done a few migrations and they're not the most exciting projects. If you do end up getting such a job, you can pick up what you need to understand what the old code did and use the appropriate equivalent in .NET.
The concepts do differ and with the rate at which new technologies are being released lately it is a step backward to ignore the new and pursue the old.
As someone who has worked with both, and with people who started with both, I'd recommend learning .Net first.
Why?
If you learn VB6 or classic ASP, you'll get used to the procedural non-object-oriented way of development. This is not a good thing, particularly if you want to move into different (newer) languages later in your career. I've worked with many people who started in procedural languages and really struggle to think in an object-oriented way. It becomes frustrating for all involved.
It's much easier to go from a .Net language to Java or C++ or even RoR or really anything that's becoming more popular.
If you're a .Net developer, you can generally write VB6 or classic ASP code. You lose a bit of what you're used to, but it's not difficult.
If you learn VB6 and classic ASP you'll probably always be able to find work. Will it be work that interests you? Maybe, maybe not. If you get into .Net though, it's an easier transition to new jobs and interesting problems and even new languages if you're so inclined. Microsoft isn't going to let .Net go, but VB6 and ASP.Net will slowly lose their support and community.
I'm going to go on a different route and say Learn VB6 and ASP Classic first and then learn ASP.NET. Learning ASP Classic first will teach you all about HTTP, HTML, and session management. You will discover what spaghetti code looks like, and you will learn how hard it is to display tabular data. With paging. Learning ASP Classic first will make easier for you to relate to PHP.
After that, learn ASP.NET and you will appreciate all the good thing it offers. I'd say that it would probably takes 1 month of small project to get yourself comfortable with ASP Classic, and another one or maybe two month to get a good grip on ASP.NET fundamentals.