If speed was what you desired in the era before trains, your best bet was catching a boat sailing to Central America where you would travel by horseback the opposite coast and then catch a boat sailing to your destination (4-6 weeks or so) which was a bit faster than sailing around South America if you managed some luck catching ships. Even in the early days of trains the Central American Transit was faster for people and light goods. There is a reason why the Panama Canal was a big deal for the United States.

Even in to the 1900's the best transit by train was multiple days (80-120 hours NYC to SF). Prior to train travel, coast to coast transit generally fell into the weeks and months category. Taking the aforementioned sea routes had some upfront expense and some amount of danger, but usually offered the most straight forward trip and if you took the route around South America, you didn't even have to leave your vehicle for the entire journey. Overland journeys usually involved a combination of roads, trails, ferrys, and boat travel.

So let's focus now on the overland routes. I'm mostly going to talk about the New York to San Fransico route as it is one of the better documentated travel routes and was probably the first/fastest route coast to coast by train when the Union Pacfic first made the link.

First lets talk about Geography. To accomplish this route a person would need to cross several East Coast rivers (ferry or bridges), cross the Eastern Mountain ranges, travese the Mid-West (and the rivers with fewer bridges a ferry crossings), cross the Great Plains, find safe passes through the Rockies, survive the difficult/hostile terrain of the Great Basin, cross the Serria Nevada Range, and cross the central valley of California. During most of the period in question, this also means crossing the territory of dozens of Native American Tribes with whom relations may have changed a great deal since you set out on your journey.

The first thing to know about such a journey is that water is your best friend but also an enemy to treat with care. Good drinking water for man and beast of burden make for important waypoints. Rivers, lakes, and streams flowing in the direction you want to travel means easier miles via boat, flowing athwart your route means finding safe passage ford or ferry provided the water isn't in flood. Logistics of needing food, shelter, general supplies, etc would generally require pack animals and wagons whenever possible. Of critical importance was procuring a good guide or scout to handle the crucial details.

Finding the best routes through the interior of the Continent was an obsession for many men going back to the early days of exploration. George Washington spent time as a surveyor plotting out counties, land, and likely finding some of the best routes to get to the better plots of land for some years before becoming a military figure. Many roads, places, trails, passes, and mountains all over the continent are named after the people who "discovered" and documented them.

The first documented coast to coast overland journey was likely the Lewis and Clark expedition (although they technically didn't start or end at the Atlantic Ocean). They set out in May of 1804 from the St. Louis area (hardly the East Coast) and returned in September of 1806. Adding on the 4-6 weeks overland to the East Coast, I suppose one could say that it took 2.5 years round trip coast to coast overland in the early 19th Century. A majority of their route was via river and speed was not their goal (finding the easiest/best route and documentation was their primary mission). Clearly 1.25 years one way is setting things at a higher end. Throughout the rest of the 19th Century leading up to the Civil War, improvements would be made to the road, trail, and rail networks that would allow fairly rapid movement East of the Mississippi River.

The Mississippi became a crucial byway for gaining access to the interior of the country as it's tributaries covered most of the midwest and navigating the Great Lakes (and traversing a few canals meant that barge/river traffic could connect a fair section of the country before the railroads increased connectivity.

Even with rail and river connections, a fair amount of time traveling from coast to coast would dictate travel via coach, stage, waggon, or horseback. The famous Oregon Trail is an example. Most folks traveling the train likely arrived to St. Louis or Kanas City by taking a riverboat or train for at least some part of their journey (though many of the travelers were already living in the Midwest before setting out).