Introductions

"Allow me to introduce you."

New to first-person conversation?  There's no better place to start than at the beginning. The preferred period method for introductions features a person presenting one friend to another--you meet people largely on the recommendation of your existing acquaintances.  

"Mrs. Jones, have you met my sister, Mrs. Smith?"

"Mrs. Denny, allow me to present M. Hugo, from Paris."

"Miss Thomas, may I present Dr. Adams?"

When making an introduction, you always present one person to the other; the higher ranking person is addressed first. To determine precedence in this and most other social situations, some guidelines:

1. Ladies outrank gentlemen (this is the most important rule)
2. Married ladies outrank unmarried ladies (also universal)
3. Older persons outrank young ones
4. Accord higher precedence to people with whom you are less familiar, and lower to your close family and friends
5. Social status—this is bit nebulous, unless you're dealing with actual aristocrats—but could come up if you're introducing two people of the similar age, gender, marital status, and proximity to yourself, in which case the one of greater social prominence goes first

Dropping in additional information (relationship to you, recent travel abroad, a visitor's hometown) is appropriate. The idea is to get people acquainted with each other, after all, and those details may help start a conversation.

Miss Leslie (The Behavior Book, 1853) writes that it is acceptable for ladies who are travelling alone to introduce themselves to genteel (female) strangers, when there is no one available to make an introduction. Similarly, persons meeting a the house of a mutual friend (according to Charles Day in Hints on Etiquette and the Usages of Society, 1844) may introduce themselves, as the host's invitation indicates that they are both respectable. Another period option is to present a letter of introduction from a mutual associate.  In certain situations (public balls),  a designated 'floor manager' or 'master of ceremonies' is delegated to introduce potential partners for the dance only; these introductions are different from normal social introductions and do not allow for future social interaction.

In any case, make sure to use social titles and surnames, when speaking to adults. Avoid reenactorisms, such as calling a middle-aged woman "Miss Betty"*, or addressing the postmaster as "George"**.  Use of first names, in period writing and address, is limited to family members, children, servants, and some intimate correspondence between friends. The exception is to differentiate between people with the same title and last name, in which case the first name is added for the junior parties.

Example: The Smith family has four daughters and three sons. Both the eldest son and the eldest daughter are married. The middle son and daughters are grown-up, but unmarried. The youngest son and daughter are still children, and not brought into adult society.

The family is addressed as:

Mr. Smith (dad)

Mrs. Smith (mom)

Mr. John Smith (eldest son)

Mrs. John Smith (daughter-in-law)

Mr. Robert Smith (younger adult son)

(Master) Michael Smith (child)

Mrs. Brown (married daughter)

Mr. Brown (son-in-law)

Miss Smith (eldest unmarried daughter)

Miss Deborah Smith (other unmarried daughter)

(Miss) Liza Smith (child)

In public, Mr. Smith refers to his wife as Mrs. Smith; she calls her husband Mr. Smith (what they say privately is their own business). They call their children by their given names. [Note: Mr. Day's book condemns the practice of calling one's own children Mr._ or Miss_, which means some people did it.]

When Master Smith grows up, he will be styled "Mr. Michael Smith" (unless/until he attains a professional distinction styling him as "Dr.", "Professor", or "Reverend", or assumes a notable military rank). The younger men will continue to use their first names in any situation where multiple "Mr. Smith"s create ambiguity.

Similarly, Mrs. Brown uses her husband's first name when more than one Brown is around. When Miss Smith marries, her sister Deborah will stop using her first name and simply be "Miss Smith". When little Liza is old enough to enter society, if any of her older sisters are still unmarried, she'll be styled Miss Elizabeth Smith. If Deborah never marries, she'll continue to hold the title of "Miss Smith"; in that case, Liza could pass right from "Miss Elizabeth Smith" to "Mrs. MarriedLastName" without ever being the sole "Miss Smith".

Women can attain professional titles, but they are very, very rare. The Rev. Miss Antoinette Brown (later Rev. Mrs. Blackwell) and her future sister-in-law, Dr. Elizabeth Blackwell, are quite exceptional--as is their shared sister-in-law, Lucy Stone.  [In order, they are the first woman ordained by a mainstream protestant sect in America, the first woman to receive a medical degree in America, and the the first prominent woman in America to keep her own name after marriage.]

References

Leslie, Eliza. The Behavior Book. Philadelphia: Willis P. Hazard, 1853. Available here. 

Day, Charles William. Hints on Etiquette and the Usages of Society. Boston: William Ticknor & Co, 1844 Available here.

Thornwell, Emily. The Ladies' Guide to Perfect Gentility. New York: Derny & Jackson,1856 Available here.

*That's "Miss Betty Thompson" if she's a spinster, with an elder sister who is also a spinster ("Miss Thompson"); exceptions if you can document a local custom appropriate to your impression

**It's only appropriate if he's your son or brother.