The government website for the General Land Office Records has been revamped, and so some of the explicit navigation guidelines here are no longer correct. I will rewrite this to cover the new step-by-steps when I can get the time. In the meantime, be aware that, although the principles and processes on some of the more complex techniques are still correct, the site navigation is different.
One big thing -- The "visitor center" no longer exists in a user-friendly way. The Reference center is more complicated for general users to navigate. The two charts that will help you to understand this system and navigate records, the layout of townships and the breakdown of land sections within townships are now on a single page (with other extraneous charts that can be ignored) at: 
The free government site, http://www.glorecords.blm.gov has records from original land purchases for all states that were surveyed by the township and range system. This includes all states except for the original thirteen colonies and those states that were originally part of those claimed lands, such as Kentucky and Tennessee, and it does not include Texas.
The 'visitor center' on the main page is very helpful, and can teach you a great deal about the how to understand land surveys and land records. You'll want to explore the visitor center in detail.
Even though only the original purchaser of the government land shows up in the database, and even though it does not cover the earliest eastern states or Texas, this website is an wonderful resource for genealogists.
The most common use of this site is to 'Search Land Patents' by name or surname in a particular state, which brings up a list of purchasers, counties and dates. You can then click on the name and get the specific land description and also print or save a copy of the original document. The help this can give is terrific!
The Next Step - Finding More Relatives
This technique has helped me find in-laws, step-brothers, maternal cousins, will witnesses, old neighbors from earlier locations, that I did not even know were in the area. These would not have been easily found by the surname search.
The technique sounds trickier than it is. It's actually quick and easy. You just need to understand how things were numbered, and the 'visitor center' at the site has nice clear help with this under 'rectangular survey system'. Be sure to print out the "diagram of a theoretical township". It really helps to have this close at hand. The sections are numbered in an odd way, but this diagram gives you not only the numbering pattern within the township but also the section numbers at the edges for adjoining townships.
Once you find a person of interest in the basic name search, and have clicked on their name, be sure to also hit the 'legal land description' tab. Jot down at least the section, township, range, and meridian. Then go back to the beginning search page where you had put in the surname. Hit the "standard" search tab.
Leave the 'names' section blank, and scroll down to the 'land description' section. Put in the section #, township # and direction, and range. Select the meridian from the drop-down list. Hit search. This will provide a list of names of others with land in that section.
Then treat your person's section as a centerpoint, and get the lists of the folks in the surrounding sections. This is where the print-out of the 'theoretical township' is really helpful. If your person's section is on a township border, get the adjoining sections in the township next door. For instance, if your person is in section 19 of the township 13N 17W, this section is on the western border of his township. His neighboring sections would be 18, 17, 20, 29 and 30 in that township. And also 13, 24 and 25 in township 13N *18* W. Sounds complicated, but your township print-out will make it easy.
And the website works smoothly. When you get your list and hit the back button, your search info is not wiped. All you have to do is change 19 to 18 and hit search, etc.
And If You Want To Go Farther...
If you want to, you can literally recreate the original plat map, with the names of purchasers, just using the land descriptions in these records. Here is where you get into the 'aliquot parts' on the land description, the odd looking things like N1\2SW or SESE. Again, the information on how these worked is available at the website's 'visitor center'. This process is a bit more complicated, but can be worth the work if you have a lot of relatives and connections in a particular part of a county.
Plat Images and Field Notes at GLO
GLO has been uploading images of the original survey plats and surveyor's field notes, available from the main page under the 'Search Surveys' tab. You can see the original drawings of a particular township, which is fascinating. Image browsing options are at the bottom of the image. Some states are not yet uploaded, but hopefully will be soon.
These plats do NOT contain any names of purchasers. But once you know where in the township your people were, you can see part of their landscape. Rivers, creeks, ridges, marshes, hills and so on. This can give you a better feel for what the land was like when it was first settled.