I knew that the expenditure of civil and political rights imposes on States, as well as economic and social rights. For instance, a State should train police or other public officials to establish a safeguard system. But I would like to know whether civil and political rights are immediate duties or progressive realization?
What duties a right imposes on government depends on exactly how the right is defined.
By definition, every right imposes a duty on someone. No duty, no right. Take "free speech." The right to speak freely means others have a duty not to shut you up, or tell you what to say. In the United States, this duty is limited to the government. As the First Amendment says:
Congress shall make no law...abridging the freedom of speech...
Among the most important duties that every right imposes is the duty of the government to provide a legal remedy, a way to legally enforce the right. If there is no remedy, then the right is just an empty promise.
Thus, the answer to your question depends on what duties the particular right imposes, including what remedies.
To see how this matters, consider the "right to an education." By itself, this claim tells us nothing about who has a duty to do what. For example, does the government have a duty to provide an education? If so, what sort of education? And "provide" how? vouchers? free government schools? Or does the right simply require that others not interfere with someone who wants to get an education? And so on.