Your question has a few issues. You asked
What is the cleanest and fastest way to handle data for a desktop application in C or C++?
Well, "handle data" is an incredibly broad description, in some sense "handling data" is all what each computer program is about. The question's body then mentions you are thinking about "saving user settings or saved games or browser history", which is a lot more specific, since now it is clear you are talking about saving and loading data, probably user specific data, to the file system which reflects the application state to make it available when the application gets restarted. I would usually recommend to do what user253751 already told you in a comment: just ask for saving user application settings (and if you have some data you don't like to call "settings", just check if you can transfer solutions to that case). But beware, if you would really do so, chances are high the question would be closed as a duplicate, see below.
The idea of a "generally fastest" approach shows a very wrong expectation. There is not "fastest" approach" outside the context of a real, specific application. How much time loading or saving of settings data takes is generally dependent on the amount of data which has to loaded and saved, and on the "portions" in which the data has to be loaded or saved. Both is totally dependent on the specific case and cannot be answered sensibly out of context. Moreover, for many applications, the performance of loading or saving user settings or other application data isn't the bottleneck worth to be optimized.
Also, there is no "generally cleanest" approach. You can evaluate "cleanlyness" for a specific program or function, but one has to see the function which saves and load the specific data and review it to tell you if there is a more "cleaner" way.
What remains in your question is a portable way of finding a base directory for storing application-specific data. The answer to this was already given in the final comment by GrandmasterB:
C++ GUI frameworks like QT certainly may offer a more universal way to get the right directory, but otherwise you have to query the OS yourself. 10 lines of code and you should be good to go.
You will find more specifics how to do this in this older SE question (for Windows, you may ignore the Registry part in the accepted answer and focus on the %APPDATA% solution mentioned here). Note this question was from 2010, where the scope of this site was much broader. Today, it is likely the wording "Best way to save application settings" could lead to quick closage by the community with "needs more focus" as well.