the Boyertown Museum of Historic Vehicles says they have the 1st car, an 1872 Hill. It was steam powered at first, and later switched to gas. Well, not exactly, James Hill, in 1868 was 13 years old and figured out the mechanics of the whole thing, and built it in 1872, though not without a lot of problems, as he was inexperienced, no one was an expert, and making his own steam engine wasn't without it's drawbacks. He swapped that steam engine out for a gas engine in 1901 http://www.fleetwoodpa.org/hillcar.html (thanks Steve!)
And, since Benz made the 1886 car powered by gas, that's why its generally acknowledged to be the 1st internal combustion engine powered vehicle made. Well, Benz wasn't the 1st automobile, but, by far the most successful at carrying on building and selling cars, and getting the historic recognition perhaps due to really high quality and great marketing.
In 1889 René Panhard and Émile Levassor entered the field independently, and the Panhard-Levassor designs of 1891–94 are of primary importance. They were true automobiles, not carriages modified for self-propulsion.
In 1897 the Winton was being sold to the public
in 1893 or 94, Haynes
In 1893 the Duryea brothers made what's considered to be the 1st car in America by most, again, though made after the 1872, it was gas powered, and they went into business selling them.
1890 or 91 William Morrison of Des Moines, Iowa builds the first successful electric automobile in the United States. http://www.pbs.org/now/shows/223/electric-car-timeline.html
in 1890 Lambert made his 3 wheel motor powered buggy. In 1891 he was driving his buggy in Ohio, hit a tree root causing the buggy to smash into a hitching post. https://skeptics.stackexchange.com/questions/14958/were-there-only-two-cars-in-ohio-in-1895-and-they-managed-to-crash-into-each-ot
And those are the earliest automobiles from 1872-1900
In March 1863 the magazine Scientific American described tests of a vehicle that weighed only 650 pounds (about 300 kg) and achieved a speed of 20 miles (30 km) per hour. Another American, Frank Curtis of Newburyport, Massachusetts, is remembered for building a personal steam carriage to the order of a Boston man who failed to meet the payment schedule, whereupon Curtis made the first recorded repossession of a motor vehicle.
However, far earlier, the steam car designers were making self propelled vehicles....
In 1808, François Isaac de Rivaz designed the first car powered by an internal combustion engine fueled by hydrogen.
I completely forgot the 1803 Trevithick goofy looking steam carriage I posted before, the one that was at Goodwood
the first post-Cugnot steam carriage appears to have been built in Amiens, France, in 1790. Steam buses were running in Paris about 1800.
Oliver Evans of Philadelphia ran an amphibious steam dredge through the streets of that city in 1805.
Less well-known were Nathan Read of Salem, Massachusetts, and Apollos Kinsley of Hartford, Connecticut, both of whom ran steam vehicles during the period 1790–1800.
the Cugnot huge, heavy, steam-powered tricycle of 1769 was said to have run for 20 minutes at 2.25 miles per hour while carrying four people and to have recuperated sufficient steam power to move again after standing for 20 minutes.