Would it be fair to say that the 14th Amendment had the effect of simplifying laws pertaining to citizenship in the U.S.A.? Can the contrasts between the nature of citizenship before and after the 14th Amendment be briefly summarized?
The primary intention of the 14th, and its largest effect, was of course to confer citizenship on blacks, particularly the former slaves. But it also constitutionalized the rule of "birthright citizenship" (aka jus soli), and removed any bars based on national origin or ethnicity.
Prior to the 14th, naturalization was a federal matter, and Congress could make what rules it pleased. But citizenship for natives was a matter of state law, or largely of unwritten common law, although after Dred Scott v. Sandford, 60 U.S. (19 How.) 393 (1857) blacks who were former slaves or whose ancestors had been slaves could not be made citizens.
The common law of England had long held that anyone born in the domains of the king was a subject, and the US followed that rule generally, in most states without ever reducing it to a statute. But state legislatures had the power to change that rule if they chose to.
In March 1790, Congress passed “An act to establish an uniform Rule of Naturalization,” which established that children born to American men abroad or at sea were still considered “natural born citizens.” It also provided rules and procedures for those who immigrated to become citizens, providing:
That any Alien being a free white person, who shall have resided within . . . the United States for the term of two years, may be admitted to become a citizen.”
Such would-be citizens had also to prove possession of “good character” and swear an oath of loyalty to the Constitution. The details were changed from time to time, but the general nature of the requirements remained unchanged all through the 18th and 19th centuries.
In the 1844 New York case Lynch v. Clarke, a court held that a daughter born in New York state to Irish parents was considered a U.S. citizen, even though her parents returned to Ireland.
[B]y the law of the United States, every person born within the dominions and allegiance of the United States, whatever were the situation of his parents, is a natural born citizen. . . . The term citizen, was used in the constitution as a word, the meaning of which was already established and well understood. The only standard which then existed . . . was the rule of the common law, and no different standard has been adopted since.
So that rule (jus soli) was not in practice changed much by the 14th, but it was made explicit and unchangeable except by later amendment. Laws on naturalization remain subject to Congressional action, and have been changed at various times.
See also this question here and its answers.