The short answer: With difficulty! The first ever traverse of the US by car was in 1903, when Horatio Nelson Jackson and Sewall K. Crocker crossed the continent in a 2 cylinder, 20 hp Winton touring car for a $50 wager, traveling from San Francisco to New York in 63 days, 12 hours, and 30 minutes. By all accounts it was a difficult and fairly treacherous journey, including such highlights as:

• Losing all their cooking gear off the car
• Losing most of their money in a coat pocket, which fell off the car.
• Losing Jackson's glasses...which also fell off the car (Are you seeing a pattern? It was a bumpy, unforgiving trip, and the Winton was completely open and noisy enough to cover the sound of cookware falling off).
• Falling off the car themselves in a crash near Buffalo, New York
• Getting lost frequently, sometimes from bad luck or directions. Sometimes this was deliberate: One woman sent them on a 108 mile detour so her relatives could see their first ever car.
• Acquiring a dog called Bud, who then had to be provided a pair of driving goggles due to the Alkali Flats near Sacramento irritating his eyes
• Horrendous roads, which necessitated replacing their tires with rope at one point due to lack of replacements. In fact, the Winton was drip-fed with spare parts sent by Jackson's wife. Ironically, one of the few parts to be replaced, the clutch cable, snapped as Jackson drove the car into his garage in Vermont on his return home.

Apparently the journey eased significantly upon reaching Omaha, due to greater availability of paved roads, but the central portion of the crossing often just consisted of them driving across the open plains or following railroad tracks, and the use of the Oregon trail saved them from a brutal Rockies crossing. The journey and the drivers became celebrities as word spread of the attempt, and even turned into a race when the Oldsmobile company dispatched a car to try to pass them and claim the glory of the first crossing.

Fast forwarding several years to brings us to the next significant crossing of the US, the 1919 Transcontinental motor convoy, a US army experiment using WW1 surplus trucks. Involving a young Dwight Eisenhower, then a Lt Colonel, the passage was a repeat of a previous 1915 attempt of the Lincoln Highway in a touring car, except done with the purpose of experimenting with long distance military logistics using trucks rather than just for publicity. The expedition was well prepared but still took 56 days to cross America, running into bad roads, deep mud and fallen bridges, among other setbacks. By all accounts it was a challenging passage, and played a large part in bringing up the need of the American military to be able to move forces quickly around the country, as well as forwarding the cause of the better roads movement, and is credited with playing an important early part in inspiring the Interstate highway program under Eisenhower in the 50's.