My question is since it would be the shortest trip to arrive in California (although I am not sure how possible it would have been) from China did Chinese immigrants typically arrive on the east coast and just take the train westward? Did they go through Mexico? or how did they get workers arriving from Asian countries to the frontier?
Chinese migrant labor in the US railroad industry was more or less pioneered by the Central Pacific Railroad, which was a corporation in practice based in Sacramento CA, and building one end of the transcontinental railroad heading east. Until the trans-continental railroad was complete Chinese railroad laborers were in effect a California only phenomenon. Most of them came directly from China by ship across the Pacific to Sacramento, and then the Central Pacific moved them by rail to the work sites in the Sierra or further east. Shipping goods and people across the Pacific had been done for centuries by the 1860s.
The reason the Central Pacific decided to employ Chinese labor and in effect recruit in China was because of a labor shortage on the west coast specifically. East Coast Americans or Europeans who took the long boat or boat and rail (across the Central American isthmus) were able to easily find work in mining or agriculture and would demand a premium in order to work on the railroad, and frequently struck for better wages. Chinese immigrants had fewer options because of rather intense racial prejudice in California. The eastern and midwestern railroads generally did not employ Chinese labor before the trans-continental railroad was complete, since they had greater access to laborers from Europe or the US east coast.