I'm not going to go into language preferences; my thoughts on this are very well recorded elsewhere.
Cross-platform is hard.  Many good programmers including myself struggle with it.  There's a lot of systems (including Python) that say they're "cross-platform", but then you end up in the "if you don't use this feature, don't call that library, and don't make this other assumption about the system you're on" land.
Pick one platform, whichever one is your primary.  Make a game for it.  I would posit that Mac is potentially a good choice from a technical perspective because if we had Core Audio everywhere else I'd not be doing Libaudioverse; Core Audio appears complex and not super well documented as far as I can tell.  XCode itself highlights every accessibility issue with the Mac, at least last I looked at it (I was, for about 2 months, going to do IOS development with Objective-c.  The slowness of getting things done with XCode stopped it).  I've heard rumors this might get better soon, but they're only rumors and definitely not substantial.
The other thing that's big here is math and basic computer science, no matter how much you want to avoid them.  Anything remotely complex is going to require both.  If you don't know much trigonometry, there are certain types of games that are basically not doable.  Namely, Shades of Doom and any other FPS.  If you aren't comfortable implementing pathfinding, you need to find a system that already does it and stay well within its constraints (I am unfamiliar enough with BGT's implementation that I do not know what that specific one lets you do or not).  The pathfinding algorithms can be done in under 100 lines, but the motivation behind why it works is not the simplest thing ever.  The alternative is to give enemies fixed paths, but this won't let them respond to changing environments like the player running away.  1d Side-scrollers are pretty easy and avoid all of this, but 2D side-scrollers like BK3 get a bit more complicated.  If you want any sort of real AI, you're headed for some non-newbie areas.  Top-down games (Shadow Rine) can be as simple or as complex as you want to make them.  I'd argue that a top-down might be easier than a side-scroller because you don't have to worry about gravity.  But really, as long as it's not an FPS, the difficulty is set by you and how much you want it to do, with the clarification that raising the bar above a certain point is like tripping on the edge of the deep end of your swimming pool while not knowing how to swim.
I can barrage you with worthwhile terms to Google, but don't think that will help at this point.  Googling is your best friend as a programmer, especially Googling for Stack Overflow answers.  Really.

Orin, what Kamlorn said above is all too true. I am a fairly experienced programmer, and cross-platform development is difficult for even an experienced coder. There are a number of languages like Java, Python, etc that claim to be cross-platform, but it only works assuming you do or don't use this or that API, library, etc. So since you are a new developer my suggestion is start with something that works on your operating system of choice first, and once you have more experience branch out to other platforms.

Regarding documentation similar to BGT for Pure Basic isn't going to happen. Let's be realistic. Audio games aren't exactly well known outside of the blind and low vision community, and Pure Basic isn't widely used by mainstream programmers. There is probably some documentation for general game development in Pure Basic, but you'd have to go through and assess what was and was not important. Probably not something you want to do being a new programmer.

To be honest documentation is one of the things I have against Pure Basic. It may be a easy enough language to learn, may work alright for developing games, but there isn't a great deal of documentation like there is for other languages. Add to that Pure Basic is not free, and you are going to end up paying out money for something when you could have tried something with more documentation and that is free to start with.

All that said I think your best option at this point might be Python 2.7. The reason I say that is Python is fairly cross-platform depending on if you use the right APIs, it is fairly easy to learn, and there is lots of general documentation out there on the language. I am sure if you ask game developers that one of us might be able to write a tutorial or sample code to help you with gaming specifics.

Honestly as a new developer you really don't need to worry about trigonometry. It's likely that you won't be at that level of programming yet for a while.

Pure basic isn't a bad language. Their is a reference manual in the help menu of pure basic that allows you to read the functions and their exampples. However, if your looking for a tutorial specificly designed to teach you how to program audio games, you are out of luck. As other's have said, stick with your current system of choice before trying to go cross platform. Before you should even consider cross platform you should get at least 1 game developed so you know the skills for developing.

The thing I found with pure basic is that finding documentation for a new programmer was difficult. there is the reference manual, but little in the way of an actual language tutorial, at least when I googled it.

If you are an entirely new programmer though, you wouldn't even know what the concept of a variable in programming or computer science was possibly. I think you had prior experience, but I could be wrong. If you have experience with a language, learning another is another thing entirely.

It is indeed true that there is no definite language tutorial for pb, but is there really a need of one? Granted, Danny got me on my feet with the language, but I'm sure that, like him, I'd probably be able to learn it on my own if I so chose to. And as for knowing what the purpose of variables is in programming and/or computer science, I'm sure it explanes it. I could be wrong, but I doubt pure basic just says, "here's how to write a string. string$ = "text" or string.s = "text"." I'm sure there's some sort of explanation of the basic concepts.

  But it's honistly your choice. I'm not like some people, who try and persuade you into their way of thinking and using their language. Like others still, I understand that people learn their own way. Like, for example, my way of learning easiest is by reference manuals such as pb's, and practicing coding. For you, stewie, that might be a language tutorial similar to the one found in bgt.

Indeed. You don't know what a variable is? Go to math class. And especially, don't start programming, until you at least know the terms. Granted, some programming languages have unique terms, beut said terms are self-explanitory, or explaned within the language's user manual.