I am a vc++ developer (but like Qt) interested in learning from open source project by contributing and reading the code. I use windows as primary development platform. Whic
Launchy
from the project page introduction
Launchy is a free windows and linux utility designed to help you forget about your start menu, the icons on your desktop, and even your file manager.
Launchy indexes the programs in your start menu and can launch your documents, project files, folders, and bookmarks with just a few keystrokes!
They also have a plugin interface so you could start with writing a plugin before getting into working on the actual code.
KDE has plenty of code in C++/Qt. It is a huge example of open source community... their SVN repository will soon have 7-digit commit numbers (http://websvn.kde.org/).
Well I think there is a lot of interesting opensource projects using c++. Boost is the first project one can think about. It is a library, but very well written, so you will learn a lot. If you want something more fun you can try something like a game engine like Ogre3D. If you want to participate in a project with Qt bindings, I suggest projects related to KDE as they mainly use Qt, AFAIK.
Boost
Ogre3D
Get involved in the Qt Creator project.
I don't think anyone here can really answer the question "Which project will be right for me to start?" as no one here really knows who you are. Programming is an absolutely HUGE area, you can do low level systems code (i.e. drivers & OSs), databasing, web browsers, games, word processing, and the list goes on.
What you should do, is find something you would like to work on, and start playing around with the code. Try out several different projects in that area and see what you can understand, contribute to and learn from (its also a good way of seeing a variety of some of the programming paradigms out there). Also, if you are trying to learn from this as well, you should check out how much help and communication between the members of the project there are and how much they seem to be willing to help one another (and if there is almost no communication, i'd worry about the project surviving much longer anyway).
If you are into gaming, try the Half Life 2.2 Full SDK. Its free and you can make your own mods.