Personally, I've always felt that Python is great for small projects and scripts, and is my own favourite language.
Like Cody said, it is very easy to learn and intuitive, and I think it looks the cleanest of all the languages (code is read more than it is written, yadda, yadda) due to the fact one would use whitespace instead of curly braces to compose blocks of code. And I just prefer that. Looks nicer, and it's easier to see the intent of the code.
On the other hand, I also like C++, because although it's 'messier' (if you contrast it with the point I made about Python) and easier for me to fuck up in, I feel I can do much larger projects with it as C++ 'feels more in tune with itself' - I think that's how I feel about compiled and interpreted languages in general. I can't really explain it any better than that, sorry.
As for projects, I jumped right in at the deep end going straight for a fully integrated GUI program - it took me ages, but I learnt/failed so much - however, a friend of mine is planning to start off with rewriting Pong and some "generic 2D platformers" (in C#/C++, I can't remember which) and is working his way towards it by doing the logic, methods, classes, etc. first, then coding a GUI in on top.
Don't know if what I said is any help at all, but hey, good luck.
P.S. I don't know any good websites that I prefer to coding on my desktop, but I think that you (or anyone) would need a proper development environment where you can do I/O, server hosting and similar stuff... and to my knowledge those are hard to find, let alone cheap/free... but, again, I've never tried hard enough to make it work.
Edited by Irradium, 06 June 2014 - 07:43 AM.