Hi, thanks for the great and fascinaring question. The answer is, as always, it depends. It depends upon a lot of factors, such as when in the 19th century you are settling the "west."

For example, the settlers who came to the Northwest Territory ( what would be Ohio, Indiana, Illinois, and such) often brought lots of tools with them. As did those who traveled the famous ( or infamous) Oregon Trail. That's not to say that they brought everything, but some basics like nails, saws, hammers, axes, and the like would have been indispensible. Settler's buiding skills also would have varied too, with neighbors relying upon each other for manpower and skillsets. Early houses were little more sophisticated than log cabins: (Harrison campaigned on being a frontier child, "born in a log cabin," long before Lincoln for example) or made out of whatever material was close to hand.

Later western expansion benefited from advances in technology and trade. Settlers to the Oregon territory in the mid- to late(ish) 19th century would enjoy supplies grom British and American (and other) traders. Whereas early settlers were mostly trappers and traders who made friends with (or married into) native tribes to have that help they seperately needed, by the 1850s and later, trade would have brought supplies, and expertise, by sea.

By the end of the 19th century, settling was not nearly so tough. As trains serviced more areas, and telegraphs made communication faster, it was entirely possible to mail order everything you needed from house plans ( complete with supplies and instructions; thank you Montgomery Ward and Sears & Roebuck!) to a bride. Granted, it took days or weeks to get what you needed, but it wasn't as onerous as in previous decades. At least you didn't have to make your "house" out of sod, canvas, and whatever else you could scrounge up!

Edit: I am on my phone and don't really have much in the way of sources close to hand so forgive the general vagueness. And spelling errors.

To expand upon /u/VetMichael has said, the exact process will vary depending upon your precise location and time period, particularly whether a railroad is nearby. In general, however, you'd be relying upon local resources, knowledge and your own tools to erect a home of your own design.

There are two particular museums that I'd recommend for this topic. The first is Kansas City's Arabia Steamboat Museum, which is devoted to a steamboat (and its cargo) recovered from the Missouri River after it sank in 1856. The Arabia was absolutely loaded with homebuilding supplies bound for 16 downs along the Missouri River. The second is the Homestead National Monument of America in Nebraska. The Homestead Act of 1862 was the legal vehicle for the white acquisition of land throughout the American West, and it defined a huge proportion of subsequent settlement west of the Mississippi River in the United States.

In the 19th century West, your tools and your skills are the most important things you'll need. Unless you're near a railroad line (and before the 1870s, those are nearly nonexistent between the Mississippi River and California), you'll only have what you brought with you. That means bringing sturdy tools, caring for them, and borrowing from neighbors (even if those neighbors are miles away). Your skills as a carpenter are critical as well, because you'll use your tools to build your home with the resources the land provides.

If you're an early settler, you're going to pick a wooded spot near a creek. Timber is an invaluable resource, and the creek will give you fresh water to drink. A deeper creek or a river might also be a transportation route, which is why you see towns and settlements spring up along rivers first.

Your timber will be milled on site; you probably will not have easy access to a sawmill, which means you'll be the one squaring and fitting the logs of your new home. You'll probably have a friend or neighbor helping you; you've probably left your children and wife back at your previous home and will bring them to your newbuilt home once it is done.

If you're one of the first people in an area, you probably won't have access to hardware and won't be carrying much more than the basics ─ probably not much in the way of nails, even. You'll use joinery and wooden pins where necessary, chink your logs with mud or similar material, and use shutters to close the place where the windows will eventually go. You'll use wooden hinges on your one front door, and in place of a doorknob, you'll use a loop of rope or wooden latch.

If you're near a river, you might have access to a store a few miles away. If the keeper doesn't have what you need, you can order it, and a steamboat will eventually drop it off for you. The storekeeper might deliver it, or simply send you a note to let you know your order is available for pickup. The Arabia for example, carried barrels full of doorknobs bound for homes just like yours.

If you're a later arrival, the streams and creeks might be claimed. You'll have to go farther away, possibly in a place where there are no trees suitable for building (or no trees at all).

In that case, you have a few options. You can use your draft animal to drag a sod cutter and physically cut bricks of turf to serve as a building material and create a soddy. You can also dig into a hillside, then enclose the resulting cave, creating a dugout home. These homes are dusty, infested by rodents and insects, and generally not your first choice, but they're cheap to construct out of the available materials.

If you arrive still later, after the railroads, you'll have access to a wealth of hardware and milled lumber ─ for a price. You'll be operating within a cash economy here, and while you could even buy a kit home from a catalog by the end of the century, that's a middle-class affectation suitable for someone who's already made it. If you're moving to virgin territory, it's not because you already have money.

If you're successful with your homesteading, if you're stable enough with your farming or herding, then you'll be ready after a few years to move out of your first home and into something more substantial, something wood-framed and shingled, pleasant to look at and live in.