Well, there was the Naturalization Act of 1790, which stipulated:

That any alien, being a free white person, who shall have resided within the limits and under the jurisdiction of the United States for the term of two years, may be admitted to become a citizen thereof, on application to any common law court of record, in any one of the states wherein he shall have resided for the term of one year at least, and making proof to the satisfaction of such court, that he is a person of good character, and taking the oath or affirmation prescribed by law, to 'support the constitution of the United States

Thus, if you're white, not a slave or indentured servant or in arrears with the law, if you're a physical resident who makes their abode within the jurisdictions outlined by this law, you go to a recognized official, take an oath, and you're a citizen of these United States.

If you're not a free white person, there's no means by which you can be officially naturalized.

Furthermore:

the children of such persons so naturalized, dwelling- within the United States, being under the age of twenty-one years at the time of such naturalization, shall also be considered as citizens of the United States.

And the children of citizens of the United States, that maybe born beyond sea, or out of the limits of the United States, shall be considered as natural born citizens

Curiously, prior to this, in 1784,Gilbert du Motier, the Marquis de la Fayette (you guessed it: French) was made a 'natural born citizen' by several US States, including in Virginia where he actually advocated for the emancipation of black slaves. (Ethnic and Racial Minorities in the U.S. Military - Alexander M. Bielakowski Ph.D)

Anyways, this is basically the law of the land for quite a while - - if you're a white person, free and in good standing with the law, the children of a citizen (even born abroad if you're not too old with respect to the law's passage, and so long as your father was a resident of the US), etc - - within pretty broad lines, you're simply a citizen.

Then came the Naturalization Act of 1795

Be it enacted by the Senate and House of Representatives of the United States of America in Congress, assembled, That any alien, being a free white person, may be admitted to become a citizen of the United States, or any of them, on the following conditions, and not otherwise:----

First. He shall have declared on oath or affirmation, before the supreme, superior, district or circuit court of some one of the states, or of the territories northwest or south of the river Ohio, or a circuit or district court of the United States, three years, at least, before his admission,that it was bona fide, his intention to become a citizen of the United States, and to renounce forever all allegiance and fidelity to any foreign prince, potentate, state or sovereignty whatever, and particularly, by name, the prince, potentate, state or sovereignty whereof such alien may, at the time, be a citizen or subject.

If you're going to be an American, you're going to be an American (and you were already white)

Secondly. He shall, at the time of his application to be admitted, declare on oath or affirmation, before some one of the courts aforesaid, that he has resided within the United States, five years at least, and within the state or territory, where such court is at the time held, one year at least; that he will support the constitution of the United States; and that he doth absolutely and entirely renounce and abjure all allegiance and fidelity

....to other foreign peoples.

Thirdly. The court admitting such alien, shall be satisfied that he has resided within the limits and under the jurisdiction of the United States five years; and it shall further appear to their satisfaction, that during that time, he has behaved as a man of a good moral character, attached to the principles of the constitution of the United States, and well disposed to the good order and happiness of the same.

Now:

0. Be white

1. Don't be a citizen of somewhere else anymore and say truly that you seek American citizenship

2. Swear you support the Constitution and American law

3. Be of good moral character and proper residence to the satisfaction of the court you're swearing in

And, a little anachronistically

Fourthly. In case the alien applying to be admitted to citizenship shall have borne any hereditary title, or been of any of the orders of nobility, in the kingdom or state from which he came, he shall, in addition to the above requisites, make an express renunciation of his title or order of nobility, in the court to which his application shall be made; which renunciation shall be recorded in the said court.

4. Badges? We don't need no stinkin' badges! No Royalty Allowed

One wonders what would've happened to the Marquis de la Fayette - - a natural born citizen of States of the United States, with his hereditary titles and free white status (and presumably of good moral character for his aid in our Revolution)

Would he had to have had settled for Gilbert?

Then came the Naturalization act of 1798

The big change with this one was that the residency requirement was now upped to fourteen years, and that one must have declared intent to become a citizen five years prior to the application.

At this time, the US was experiencing a wave of immigration from France and Ireland, and those immigrants tended to support the politics of Thomas Jefferson and the Democrat-Republican party which promoted decentralized government and States Rights (amusingly a merger of the "Anti-Administration" and "Anti-Federalist" Parties) which was in opposition to the still not very creatively named Federalist Party which supported a stronger Federal government. Better than just anti-The Other Guys, but, still, come on.

States still set voting requirements (and in part, they still do to this day despite some later legislation in the Civil Rights era), so immigrants could vote (without being citizens!) but along with this Naturalization Act, the Alien and Sedition Acts of the same year prevented immigrants from readily becoming citizens and holding Federal office or appointments. It seems to be pretty well accepted that these laws were pushed for by the Federalist party in opposition to Jefferson and his ilk becoming potentially more powerful (Encyclopedia of American Civil Liberties - Paul Finkelman pp 37), in addition to there being worries about war with France - - - they, of course, had had a revolution of their own, and there were naval conflicts between the US and 'New France' that made a lot of American lawmakers worried about full blown war.

Side note:

This literally pitted us against the French (right after they helped us get the British to surrender) and alongside the British. War's weird.

Side side note:

It seems when Jefferson became president, the 1798 law not favoring his party was subject to some executive discretion and its enforcement was deferred....I'm not going to wade any further into that water, but make of it what you will that one party sought restrictions on immigration because it meant new foreign voters would make them less powerful, and then the president of the opposing party suspended the enforcement of those laws because....

Anyways:

Enter the 1802 Naturalization Act

This was pretty much the last thing we passed until the 1860s when Dredd Scott's citizenship became an issue, but it more or less just created registries at all kinds of district levels so that we had records of citizens, but the basic stuff was all pretty much the same as prior laws.

Essentially:

We've always had a process by which people can become citizens, we've always had political fights over who gets to be a citizen and how and why (motivated by political gains), and it wasn't always very equitable under the very principles of the Republic our nation fought for in the Revolution.

As a follow up question, how did one prove he was a citizen before the passage of the 14th amendment?