If I was the valedictorian of my high school and wanted to attend Harvard in Massachusetts, would I just send them my application with grades and everything through the mail

Yes, and this is still how the vast majority of people get accepted to schools.

For years after airplanes were invented, the most popular ways to travel around the US were: boat, train, bus and car.

Stagecoaches weren't as popular for long distance travel, unless you were rich. By the mid 19th century, you would probably take a train. Still, going from west to east could take 4-5 weeks. You have to understand, people were just not as mobile back then. Sure, many people made the trip, but the vast majority did not.

Anyway, even during WWII, trains were still the main form of transportation for servicemen. To travel around the country privately, you would take things like Greyhound, which had over 10,000 employees and almost 5,000 stations when the war began, but as a solider you'd only travel to the nearest depot to meet up with fellow recruits and then get put on a train.

Really, the first war you see where US soldiers traveled around the country to stage for deployment via airplane would be the first Gulf War in 1991.

Rail and boat were the kings before airplanes. California to the east coast was really tricky because you had to go around South America by boat and sailing off the west coast isn't particularly fun even today. So a rail line to there was a very big deal in US history.

If you can get to someplace by rail that'll be better than by boat. It's reliable, more comfortable, more cost effective and certainly faster. Prior to rail if you couldn't get somewhere by boat, then it'd be wagon or coach. It'd be a miserable way to travel.

Because it seems that in these days, many Americans turn 18 and either go to college or join the military.

But this wasn't the normal thing to do before WW2. Going to college was for a wealthy or brilliant minority. Most young men went to work, and so did young women of the blue-collar class. Many were out of school and working at 12. The farther back you go the worse it gets. Joining the Army was a very tough low-class job that made you travel a lot. A man needed an adventurous streak and no better prospects to join the ranks.

From 1840 forward, trains rule wherever they go. The 1904 2000 Miles on an Automobile notes that outside cities and villages roads have decayed to rough tracks because no one drives a wagon any distance: people and cargo go by rail. The book has a lot of hilarious incidents of getting stuck in mud or sand, or finding there's no road to speak of. That's why building the transcontinental Railroad was so important after the ACW.

Before then, California was not "a big population center." Most of its population came for the 1849 Gold Rush. They came by wagon train and on horseback overland, a miserable trip through wilderness once they hit the Great Basin. Most came by sailing ship from the East Coast, down the length of the continents. "Doubling the Horn" was often a matter of waiting and fighting gales, still, and then sailing up the western coasts (catch Two Years Before the Mast for a later cargo voyage).

So the son of a wealthy Californio who did go to Harvard in 1855 would probably sail east. He might elect to shorten the trip by sailing to a Mexican port, going overland a shorter route, then sailing from Vera Cruz north. This had its own overland problems.

Stagecoaches were for places the rails hadn't reached yet. They were dusty, and rattled over very poor excuses for roads. There was often no alternative, but if you weren't in a hurry and you were in good condition, it was more comfortable to ride a horse overland.