The Civil War is beginning, of course. And the Homestead Act of 1862 hasn't been passed. So where are people who are going (further, as I'm including those already pretty far west) west going, especially if they are hoping to avoid actual fighting? Especially, are there any not-too-built-up/expensive areas that American Indians have already been driven out of - for those that want to avoid that kind of fighting, too.
Exploiting the Homestead Act - even though it postdates your wish to flee - would probably not have removed you from the threat of being drafted and fighting. The Homestead act was best used for agricultural purposes and during the Civil War, the Midwest was too close not to contribute its share of cannon fodder.
Your best best is to try something other than agriculture. California was far enough removed that it was not looked to for recruits, so that's a good bet. You might try your hand at farming, but by 1861, the land was being locked up and was increasingly expensive. There were, however, plenty of opportunities for employment or for establishing a business, so I recommend that.
Similarly, Oregon and Washington are possibilities, and there, it would be possible to acquire land that good for agriculture. The dispossession of Native Americans was being horribly executed at that time, however, so you would likely have to be part of that process. I don't recommend it.
Many in 1861 were heading to the Intermountain West and its dozens of mining districts that were being established. That part of the West tended to be highly urbanized since the land was poor for conventional agriculture. It could sustain widely scattered herds of cattle and sheep, but that meant the population (and opportunities) were widely spread and relatively rare. Best to stick with the cities.
When President Lincoln appointed Orion Clemens as secretary of the Nevada Territory in 1861, his kid brother, Samuel Clemens (1835-1910), tagged along. He was clearly trying to avoid the conflict, and in part he was motivated by a fear that Union forces would impress him to serve as a Mississippi riverboat pilot. Mostly, he didn't want to be shot, so going to the Nevada Territory was appealing, and he made his way by settling in one mining camp after the next. After taking the name, Mark Twain while working as a reporter in Virginia City, he was forced to flee in May 1864 - he then went to San Francisco for the remainder of the war. Clemens/Twain serves as a good example of what one could do in 1861 if one had an aversion to lead poisoning.