Travelling from New York to California before the transcontinental railroad opened in 1869 would be long, expensive, and dangerous. But going by ship would be the best way to go.
Most people who moved to California overland joined a wagon train in western Missouri
The California Trail was an emigrant trail of about 3,000 mi (4,800 km) across the western ...
Because it was faster and much safer.
Traveling by coach wasn't a thing. Much more going with some pionieer wagons. We know that it took them around 4-5 months. Beetwen 120-170 days (depending on where in California).
Pony Express (or lone horse riders) - in 1861 they claimed that it will take 10 days to deliver letters from New York to Sacramento. Small ...
The movie takes place in 1862 (according to wikipedia), and the transcontinental railroad was started in 1863, so the alternative is something like the oregon trail. Jenny isn't a pioneer, just a traveler. On a boat, things like food and shelter would be mostly taken care of.
Also, I suspect it's a bit more romantic, so makes for a better movie