QuickBasic will give you a huge assortment of general-purpose flexible programming skills for anything from working with databases to 3D graphics engines.
That is, if you learn most of the "intermediate" stuff QB has to offer - I mainly have in mind:
-Arrays
-Working with sequential and random-access files
-Subprocedures (SUBs and FUNCTIONs)
Graphics (PSET mainly) also, though that would be more as practising all the other skills than learning anything big.
Basically, when you know how to program, learning a new language is much easier. And the more practice you get, the better you'll be able to program.
I'm not saying QB will teach you everything you'll need to know, but it CAN teach you how to think like a programmer, how to come up with a method for solving a problem, and how to look for information. I very, very strongly recommend you get familiar with QB help.
Because learning programming isn't about learning all the commands in a language - not when there are thousands of them and you can add your own. It's about deciding roughly what commands you will need, and then being able to find the commands and how they work.
And (unlike learning all the commands), this is a skill that is useful any programming language.
To more directly answer your question:
Get fairly deep into QB. It's easy to learn and fun (especially when compared with most other languages) but gives you great foundation skills. For example, learn to sort arrays alphabetically, search for something in a file, make a ball bounce around the screen.
What next depends on where you want to go. "Modern programming" says not much. You could work with databases, user interfaces, game engines... (You'll probably have a better idea about this once you've had more experience.)
Personally I'd learn C because it's logical, challenging, and lets you produce practically the most efficient software of any language. Then I'd learn either C++ or Java to get used to the "object oriented" concept. If you do all that, you can work practically anywhere, because once you know everything that BASIC has, as well as pointers (mainly C/C++) and OO (a lot of "modern" languages use this to some degree), you have almost all the concepts you'll need to use loads of languages including ASP, C#, PHP, Delphi... All you need to do is learn their syntax and have a reference handy.
But you may not be up for it. Maybe you just want to specialise in databases or graphics or something. You may also want to focus on the .NET languages which will probably be useful for internal business programs (but not for commercial software).
Here once again is my tutorial though it's still unfinished.
aleks.budzynowski.info/wiki/quickbasic/tutorial/intro