Are you able to get an antibiotic (e.g. Amoxicillin, common/general Antibiotics, etc.) from a pharmacy, with no doctor visit required, while traveling through the major cities in Russia?
As others have already noted, formally many drugs are prescription-only. However, I never met a pharmacy that would insist on you showing prescription, though I buy something regularly (say, once or twice per month).
In 99% cases, the staff will ask nothing, and just hand over the drug needed. I remember only once being asked for prescription, and even then a simple "I forgot it" worked. The only cases when you might encounter some problems are if you raise suspicion (for example, the drug is really unusual), or if the drug it can be used to create some prohibited addiction/narcotic drug ("get high", as user28434 put it). The latter is because sale of such drugs is tightly controlled by narcotic/drug control police, but they are interested only in sale of some very specific drugs, not general antibiotics for example.
I mostly visit pharmacies in my home city, Nizhny Novgorod, so this may be specific to this city. However, several times I also have been buying drugs in other places, including large cities such as Moscow or St. Petersburg, medium-sized cities as Gelendzhik, and rural area of Kostroma oblast, and everywhere the attitude was the same.
De jure no drug is purchasable without medical prescription in Russia, because the law that listed such drugs was canceled in 2011.
But, de facto, pharmacy staff can decide if you can buy it without a prescription. So it depends; if the drug can be used to get high, or if you're buying it in huge quantities then you’re not likely to be able to obtain them.
It really depends on drug you want to buy. For example, drugs containing Paracetamol, Aspirin, Analgin are most popular non-prescription ones so you will not have any problems to purchase it. Maybe that will be enough for you, and won’t need to buy the antibiotics for your wife.
However, drugs you've mentioned such as Amoxicillin are prescription only, and it can be a challenge for you to get that one. As @user28434 said, antibiotics can be purchased only with prescription, it's a law.