There are twenty-two Georgia matches for W Farmer. The same man can and often will be listed more than once in the index, depending on how many different ways his name is given in a service file. The same soldier could have served in more than one Georgia military unit, so that's another source of duplication.
When researching a name like this, I prefer to indentify all male residents living in the state whose name may fit. The search can be narrowed to include those born between 1848 and 1808, and further eliminate men listed in certain professions such as ministers, physicians, engineers &c. What's left are names that should be reasonable matches to index entries.
The index will give company and regiment for each name. Once you've listed these, eliminate nearly identical names and begin looking for county of origin for each company on your list. Targeted searches on the internet should help with most of these. Remember that most Georgia counties are relatively small, so men from one county often enlisted over the line in a neighboring county. You should be able to accomplish all of this and narrow your focus to just a few service files before even thinking about going to the state archives. It would also make the visit much more productive.
At this point you can start to look for additional details from pension applications, service records or even service record summaries. For instance, if you are certain your ancestor survived the war, you're probably not interested in soldier who lived through it. Pension applications can be a great help if you know where your ancestor or his widow lived after the war. Bear in mind that many veterans never applied for state support, that pension legislation wasn't enacted until many years after the war, and that only veterans and their unremarried widows could apply.
Most people are able to select one name from the list and order a service file and/or pension application for him. You can order those here. As mentioned, many soldiers served in more than one company, so the message board offers a discounted rate whenever orders for more than one file are received.
Let us know if you have other questions.