Where did sailors in the 1700's-1800's get tattooed, what was the tattooing process, and afterccare like?

The best article I know of concerning sailors and tattoos is this article:

Dye, Ira. “The Tattoos of Early American Seafarers , 1796-1818.” Proceedings of the American Philosophical Society 133, no. 4 (1989): 520–554.

If you're thinking when you say "where" that there was a tattoo parlor/shop they would go to, as far as I can tell, there weren't in the eighteenth century and early nineteenth century. So you're answer to "where" would be wherever it could be done. Here are some pull quotes relevant to your question:

"From the sketches and descriptions...it is probable that most of the tattoos used in this time period were line drawings in black or blue, or occasionally in red, executed by sailor-amateur tattooists."
"These sailor-amateur tattooists could have been professional or semi- professional in the sense that they were skilled and charged for their work, but they were probably also still seagoing sailors who tattooed their shipmates, rather than full-time expert tattooists similar to those in the tattoo parlors of today's seaport cities with their plastic patterns, electric needles, and multi-colored inks."
"The design was traced on the person to be tattooed with a pen or pencil. Coloring materials, apparently nearly always Indian or Chinese Ink (lampblack mixed with animal glue, sold in solid rolls or cakes), laundry bluing, or vermilion (artificial cinnabar, i.e., alpha mercuric sulfide, ground with white wine and then mixed with white of egg), were prepared with a little water or saliva in small cups or shells. The tattooing instrument was formed by tying two, three or four needles together. The tattooist stretched the subject's skin as tight as possible in the area to be worked on, then dipped the needle bundle into the coloring matter and, holding it at right angles to the skin surface, pricked the points deeply enough to carry the coloring matter into the dermis, below the epidermis. This was repeated until the design was completed or the subject could no longer stand the pain. After the session was completed, the blood and excess coloring matter was washed off the subject's skin, sometimes with water, sometimes with urine (probably in the belief that it served as a fixative), and sometimes with rum or brandy. Any rum or brandy left over was apparently a perquisite of the tattooist. Sometimes the coloring matter used was wet gunpowder and ink, which would be rubbed into the needle marks."
"To modern eyes, these practices appear dangerously unsanitary. While there is no description of an American seafarer of the period suffering serious infection or disease from tattooing, there is plenty of evidence that others who were tattooed by using the common process of that time and later were afflicted with gangrene, syphilis, leprosy, and/or viral hepatitis, among other things. A modern dermatologist discussing the complications that can arise from even today's more "sanitary" tattooing lists twenty-seven conditions."
"Available evidence indicates that the professional tattooist came on the scene in American (and British) seaports around 1860-1870."

Any more detail than that, including designs of the period, and I would highly advise acquiring the previously referenced article yourself for further study.