1) You'll likely get better/more response if you prefix your post with "Ask HN: " (lots of folks have triggers for that type of thing)
2) You don't have to be a mathematician to be a good programmer. Tons of great programmers were not comp-sci majors so don't let that sort of thing get you down. If you've done "some basic noodling here and there... in a few languages" and you want to improve as a programmer, I'd advise you to:
- Pick a language
- Pick a pet project
- Buy a few books or take a few online classes that are on your chosen language
- Begin reading the book
- Implement your project while reading the book
- Find some sample apps written in your language and review how they did things differently
- Finish your book and buy another book/class that is specific to your language
- Pick a new pet project and implement it
- Review some open source apps written in your language
- Contribute to some open source apps
* The Pragmatic Programmer
* Code Complete
* Head First Design Patterns
* The Mythical Man-Month
* Design Patterns: Elements of Reusable Object-Oriented Software
Next up: build a startup!
Notice there are no math books in my list. This is different from what others would propose but, given the scant description you wrote, I don't see any need for math except to satisfy a possible desire to learn.
I am really interested in investigating at a deeper level so I can think in a more mathematical way about programming problems. Efficiency, the basic algorithms you should know, and so on. I will look into those books you suggested. I haven't read any of them! Again, thanks for taking the time to respond. Then, perhaps, a startup, as you say!
Best of luck to you
Its a good introduction, but it will require you to be decent at math and know how to do proofs.
On the other hand, if you are interested in how programs can express things, then maybe you want to learn about some CS theory. Specifically, some lambda calculus would be good to learn, but I don't have any good suggestions.
Note: All of these will be easier to read if you are pretty good at programming, so I'd say follow at least the first piece of advise above.
Beyond that sort of thing, you need to specialise in the kind of areas you want to work in - if you want to program to earn money, then learning SQL, HTML, and web app development in your favourite platform is a good idea. If you go for the Unix route, I'd recommend you learn Python before Perl/PHP, but then learn the latter two because there's money in it. If you go the Microsoft route, I'm sure there's plenty of good books on C# and ASP.net. Java's pretty useful, too.
It's a good idea to learn C, not so much to learn the language itself as to learn about the processor's memory model, which C somewhat ruthlessly exposes you to; if you master C, you'll indirectly learn how software performance works, in ways that'll indirectly help you write faster code in higher-level languages. And, of course, C is useful in its own right if you want to get into embedded, systems, or games programming.
Personally, I think C++ isn't a good language to get involved in; if you want to anyway, either because you disagree with me or because you're after a job doing C++, make sure you've read Structure and Interpretation of Computer Programs, played with C, and tried at least one object-oriented language first; C++ is complex, and having a strong background in the basics behind it will help you a lot; trying to dive straight in might lead to a rather painful learning experience.
Perhaps most importantly, find a programming project you are excited about, and do it. When you're done, you'll have realised you'll have made lots of mistakes - but you'll never make those mistakes again, and the next project you do (perhaps one you'll be paid for) will be a lot better. Practice makes perfect! And have fun!