Alright... I had to look some up and I wanted to know:
Chicago Cubs - 1902, established in 1870 as the White Stockings, renamed the Colts in 1890, Orphans in 1898.
Atlanta Braves - 1966, established in 1871 as the Boston Red Stockings. Name changes: Red Caps 1876, Beaneaters 1883, Doves 1907, Rustlers 1911, Braves 1912, Bees 1936, Braves 1941. Moved to Milwaukee in 1953 then Atlanta in 1966.
San Francisco Giants - 1958, established in 1883 as the New York Gothams, changed name to 1885 to the Giants.
Los Angeles Dodgers - 1958, established in 1883 as the Brooklyn Atlantics. Name changes: Grays 1885, Bridegrooms 1888 and 1896, Grooms 1891, Superbas 1899, Trolley Dodgers 1911, Dodgers 1913, Robins 1914, Dodgers 1932.
Pittsburgh Pirates - 1891, established in 1882 as Alleghenys.
St. Louis Cardinals - 1900, established in 1882 as the Browns, renamed Perfectos in 1899.
Cincinnati Reds - 1958, established in 1882 as the Red Stockings, renamed Reds in 1890 and briefly changed their name to Redlegs in 1953 because of McCarthyism.
Philadelphia Phillies - 1884 (occasionally called the Quakers and they unofficially tried to change their name to the Blue Jays at some point).
Baltimore Orioles - 1954, established in 1894 as the Milwaukee Brewers, moved to St. Louis in 1902 where they were the St. Louis Browns.
Chicago White Sox - 1904, established in 1894 as the Sioux City Cornhuskers, then in 1895 they were the St. Paul Saints, in 1900 they had no "official" name but were known as the White Stockings.
Cleveland Indians - 1915, established in 1894 as the Grand Rapid Rustlers, named the Cleveland Blues in 1900, named Cleveland Naps in 1903.
Minnesota Twins - 1961, established in 1894 as the Kansas City Blues, named the Washington Senators in 1901.
Boston Red Sox - 1908, established in 1901 as the Boston Americans.
Detroit Tigers - 1901
New York Yankees - 1913, established in 1901 as the Baltimore Orioles, in 1903 they moved to New York and called themselves the Highlanders.
Oakland Athletics - 1968, established in 1901 as the Philladelphia Athletics, moved to Kansas City in 1955.
Los Angeles Angels - 1961... changed city name too many times for silly reasons.
Houston Astros - 1965, established in 1962 as the Colt 45s (changed name because of the astrodome).
New York Mets - 1962
Washington Nationals - 2005, established in 1969 as the Montreal Expos.
Milwaukee Brewers - 1970, established in 1969 as the Seattle Pilots.
Texas Rangers - 1972, established in 1961 as the Washington Senators.
Kansas City Royals - 1969
San Diego Padres - 1969
Seattle Mariners - 1977
Toronto Blue Jays - 1977
Florida Marlins - 1993
Colorado Rockies- 1993
Arizona Diamondbacks - 1998
Tampa Bay Rays - 1998, changed name from Devil Rays in 2008