How did archeology get its start?

It developed from the antiquarians of the late nineteenth century. It is not something you can give a start date to. Though what resembles the modern idea of Archaeology begins to appear very late nineteenth / early twentieth century.

The history of archaeology as a science begins in 18 century and is connected with famous English scientist Peter Reverse, He was the first who said that the main point of archaeology is not Golden things, but the piece of pottery may be much more important.