Period authors, philologists and living history reenactors represent just a few people who require a reliable understanding of speaking 18th-century antiquated English. With so much happening in the Anglo-American world during this time--pirates on the Seven Seas, Enlightenment philosophy, the French and Indian War and the American Revolution to name a few--understanding the speech patterns of the 18th century is particularly useful to many people. Fortunately, English spoken in the 1700s is quite similar to what is spoken today. You're not really learning a new language, you're just picking up a few quirks.
Model your pronunciation after a subtle, class-neutral London accent. According to Craig Carver, around 1400 the English language evolved into this form through what philologists call "The Great Vowel Shift." Because most English printing presses were located in London, the dialect of that area became standardized and came to define the language as a whole. Other dialects gradually developed from the blending of native and immigrant populations and so were not as prevalent in the 1700s.
Read a wide assortment of English works written in the 1700s encompassing several subjects and audiences. Novels and political essays are good resources to learn polished speech of the day, while stage plays often present the language of the common man.
Study the proper use of English pronouns. A very common misunderstanding in modern readers of period literature is the difference between "thou/thee" and "you/ye." During the 18th century, "thou"and "thee" were the second-person singular familiar pronouns, meaning that they served to mean "you" or "yourself" EXCEPT when people of high respect or multiple people were being addressed. "You" and "ye" were used only when multiple people or respected figures were being spoken to. "Thou" and "you" are used as the subject of a sentence while "thee" and "ye" are used as direct or indirect objects. For example, "I shall give thee what thou shalt want" means "I will give you what you will want."
Study early modern English verb conjugation. While there are many exceptions to the rules, 18th-century English commonly employed a "-th" ending for verbs used with third-person singular subjects and "-est" endings for "thou" and "thee." For example, "Thou knowest that he runneth," means "You know that he runs."
Pepper your speech with words that have evolved or gone extinct by reading contemporary historical English lexicons. It was acceptable for authors to invent words with Greek or Latin roots, so they employed words that have completely changed in meaning or simply never caught on. An invaluable resource is a dictionary that provides thorough etymologies, as these often tell the convoluted stories of how a word got its meaning.
English orthography was not yet standardized, so the same word may have had several spellings at any given time in the 18th century. This can often cause confusion for later readers.