Are you sure about that? I may be wrong of course, but I remember pressing "F3" (which I think was meant to invoke help?) and getting nothing. That's actually why I bought a book, I think this one: http://www.amazon.com/Crash-Course-Qbasic-Greg-Perry/dp/1565... (quite a nice one! I remember reading it cover-to-cover during summer break on a beach, without any computer in at least 20km radius...)
But it's absolutely possible that I either had some ripped version or just couldn't find the right options. I mean, I was 8 at the time, knowing very little English, so that wouldn't surprise me at all :)
Python comes close. It runs on just about anything, has a very basic GUI toolkit in the standard library (Tkinter) and comes with a syntax highlighting code editor with built-in command shell as standard (IDLE).
Tkinter is ok for simple apps like a calculator. IDLE was written in it so it's fine for basic apps. For anything more complex there are wrappers available for native UI libraries, or you can use PyQt or PySide that both wrap the very capable Qt framework (a personal favourite). There are more feature complete IDEs, but honestly IDLE is a fine learning environment.
There is a commercial version of Python available for the iPad called Pythonista. It has a serviceable code editor and a bunch of external libraries baked in, but doesn't have TkInter as the desktop GUI paradigm doesn't work on iOS. Instead it has wrappers for the native iOS GUI frameworks, and even comes with a visual UI designer.
There's also NetBeans, KDevelop, Komodo, Virtual Studio Express, Xojo, and more. It depends what you want to do.
And if you're on a Mac, I'd go with Xcode+Swift. Great language for beginners.
Still, bundling a quite capable IDE with an OS was a very nice practice, but it basically ended then, probably because MS also sold Visual Basic. Without QBasic I wouldn't have started programming at all - I wonder how it would be if I was starting right now. Is there any ubiquitous, simple language available basically everywhere and with a nice IDE by default? JavaScript almost fits the bill, but as a beginner who doesn't know any better you're likely to use Notepad, while with QBasic you had a real IDE from the beginning.