It depends on when you were going to try to do it. Horatio Nelson Jackson was the first person to travel coast to coast by automobile, when he set out from San Francisco on May 19, 1903. He arrived in New York City on July 26, which was 63 days, 12 hours, and 30 minutes later, covering a distance of 5600 miles on the journey. It's estimated there was only 150 miles of paved roads in the U.S. at the time (and most of that would have been in cities).

In The Record-Setting Trips: By Auto from Coast to Coast, 1909-1916 by Curt McConnell, the author details other early coast-to-coast drives. Among them were Alice Ramsey's first cross-country trip by a woman, which took 59 days in 1909. A 1916 race cut this time down drastically. Edwin G. "Cannon Ball" Baker won a New York to Los Angles race that year in 7 days, 11 hours, and 52 minutes covering a total of 3,381 miles.

In 1926, the federal government enacted the U.S. Numbered Highway System, and began efforts to upgrade the highway. By 1927, it was possible to get from Los Angeles to Washington, D.C., relatively easily, by following U.S. Route 66 from Los Angeles to St. Louis, and then from there, following U.S. 40 (the old National Road) from St. Louis to Washington, D.C. Ernest N. Smith, the head of the American Automobile Association, covered the route that year, his sixth cross country trip (though not all by this route). The Route 66/Route 40 trip took Smith 39 days...but he was giving speeches, attending conferences, and meeting with local AAA chapters along the way, on and off the route. He actually traveled 9000 miles in those 39 days, averaging about 225 miles a day. Had he gone the route from end to end without detours, which was about 3300 miles, it probably would have taken him about 14 days, assuming he was still only aiming for the 225 miles each day.

But if his aim had been to cover as much mileage as possible, he could have done better. In 1933, "Cannon Ball" Baker set a new record. Again, he raced from New York to L.A., as he had done in 1916, but this time he could follow the U.S. 40 and U.S. 66 routes. For this race, he set a solo record that stood for almost forty years: 53.5 hours, sleeping for only a half hour along the way.

While it's difficult to give an exact estimate, taking into account "Cannon Ball" Baker's records, and Ernest N. Smith's more leisurely cross-country trips, it's reasonable to assume that, from 1927 after, multiple drivers, in a brand new, reliable car without catastrophic auto failures, and with a full gas can on board and other provisions in case of emergencies, could probably have done the Route 66 --> Route 40 trip in about a week to ten days. Just doubling Smith's daily mileage by using two drivers, Smith's mileage could be covered in 12 hours of daily driving averaging 37.5 mph, and the trip would be accomplished in about 7 days and a few hours. That leaves 12 hours each day for stopping, sleeping, and eating, and making up for lost time due to any delays.

With those same drivers driving nonstop, it may have been possible to make the trip in as little as 3 to 5 days in ideal conditions.

Between the mid-1910s and 1927, before the U.S. Highway System made the trip easier, it would have taken longer. Under non-race conditions, it probably would have been wise to set aside at least two or three weeks, even with multiple drivers with the aim of driving non-stop. Hazardous road conditions, impassable roads (there were points where it would require your car to be ferried across waterways), the top speed of your automobile, and the inevitable mechanical issues would have made it unlikely that you were going to complete the trip without significant delays.

Before the mid-1910s, with barely any paved roads at all, and in an even more unreliable and slower automobile that might not even have had headlights, and without gas stations and other roadside conveniences, you probably would have wanted to set aside at least three months to make the trip.