How did archaeology start as a field of discipline?

It started nearly 500 years ago as an interest to the different kinds of objects.
1533- King Henry VIII, hired first "archaeologist" as antiquities clerk.
1612- First "archaeological" excavation at Rome by Howard.
1690- First Paleolithic artifact found in London. It was a biface.
1718- First NGO about history: Society of Antiquaries (London).
1738- Excavations at Pompeii and Herculaneum.
1744- First excavations in northern Europe: Zealand tumulus in Denmark.
1759- British Museum founded.
1784- First excavations in north America: President Jefferson excavated Monticello.
1797- First correlation between Paleolithic stone tools and animal bones.
1799- Rosetta Stone discovered.
1817- First systematic exhibition: Thompsen arranged Danish museum according to his "three age system" which is still in use (Stone, Bronze and Iron ages).
1833- Stratigraphy rules by Lyell about geology.
1839- Discoveries of Mayan cities in central America.
1842- First systematic Paleolithic excavation: Abbeville.
1854- First ziggurat excavation at Mesopotamia: Ur.
1856- Neanderthal man identified.
1859- Darwin published Origin of Species.
1865- First appearance of word "Prehistory" by John Lubbock.
1879- Paleolithic paintings at Altamira Cave discovered.
1884-1908- Excavations by Schilemann (at Tiryns), Vesselovskii (at Maikop), Evans (at Knossos) and Vasic (at Vinca).
1921- Homo erectus fossils found.
1922- Grave of Tutankhamon opened by Carter.
1922- Sumerian civilization to be popularized
1924- Discoveries of first great civilizations in India.
1925- First published synthesis about prehistory by Gordon Childe: The Dawn of European Civilization.
1926- Leakey started researching in eastern Africa.
1948- Braidwood's Jarmo excavations make nature scientists and archaeologists work together.
1949- Radiocarbon (C-14) dating invented and used to date archaeological materials by Libby.
1952- First underwater excavations by Cousteau.
1959- Olduwai finds in Africa.
1974- Chinese terra cota army discovered.
1991- Ice man "Otzi" discovered in Alps.
1994- Chauvet Cave discovered.

Archaeology started as a research for roots of Middle age civilization in the Roman and Greek civilizations. It's the beginning of Classical Archaeology. European researchers started to search the holly places that are written in the Bible. This is the beginning of Near Eastern archaeology. With the Industrial Revolution, geology started to develop and it is the beginning of prehistoric archaeology. With the 20th century, every national state in Europe started to search their roots. So archaeology became more complicated and sponsored by the governments.
Source(s):

ArkeoAtlas 1 (2002): 28.