As Cyclops mentioned, Unity doesn't use C++, it uses C#. Since you are new to programming C# would be a good jumping off point.
Personally, I started by picking up a book and programming on a TI99-4A way back in the day (external cassette backup...which never worked properly). There are a number of good books to teach-yourself programming basics. Here's a couple of listings off of Amazon (C++ and C#).
Some of these books will include a CD with a compiler, samples, and examples. If they don't, here's a free compiler:http://www.mono-project.com/CSharp_Compiler
Avoid "for dummies" programming books...I think they are too simple. Avoid what is called "visual-c++, visual-c#,..." those normally teach a person more so on programming a gui, not to be confused with scripting a gui as with Unity.
The things you will need to concentrate on are: creating functions, passing variables, and class structs. (and of course syntax)
One last thing to add, you mentioned that a lot of developers still use C++. Yes and No. From my experience they use C, C++, and C# and a lot of custom libaries. The regular Windows libaries can be really bloated.
Just my 2 cents...and my first posting on Unity.