It is time for me to get a new phone. I will be going to Ukraine in a couple of weeks.
I know absolutely nothing about how cell phones work. I know the term sim card, but have no idea what it means.
What is a sim card?
It is the "brain" of your phone. It is issued by the cell phone carrier (e.g. Vodafone, Telekom Ukraine, MTS) and connects your phone to their network. There are many different carriers and plans, so it makes sense to research what you want and need before you buy one. In Ukraine and Russia you can "top up" your pre-paid SIM card at kiosks EVERYWHERE, so that you don't run out of credit.
My question is this: Is there a phone I can purchase here in the US that works in both places?
Yes. Ask your local cellphone dealer for an unlocked quad-band phone, or buy one through Amazon, e-Bay or similar. This should cover the 850 MHz, 900 MHz, 1800 MHz and 2100 MHz bands (although you probably won't need the 850 coverage). These are the frequencies used for GSM (2G) and UMTS (3G) coverage in Europe. You should be able to get one for under $100. "Unlocked" simply means that the phone is not tied to a specific network provider (e.g. AT & T), but will work with any carrier.
Does it make more sense to buy a phone in Ukraine?
Probably not, because it won't necessarily be as good as one you buy in the USA, or may be more expensive. I bought a new cheapie in Moscow after my own phone was stolen in China, for the equivalent of $30 US. The staff at my hotel reset the phone language to English, and I haven't had a problem with it. I gave it to my mother when I returned to New Zealand, and the only (very minor) problem is that it has to be charged using a European power adaptor (which she already had after travelling there). It works fine with a New Zealand SIM card.
Would the same phone in Ukraine work in other FSU countries?
Yes. It will work anywhere in the world if you have the appropriate SIM card (see above).
Phone preferences for me are pretty simple. I have little to no interest in apps, it might be nice to have a good camera, I generally don't care for Apple products, just don't like the proprietary nature of needing everything to be different just because it is an Apple. (But if the Iphone is really the best option I would buy one).
Me too, so I don't have a smart phone. My phone is for phone calls (funny that!) and occasional texts. Even though it has a camera I would rather use the digital one which I always carry with me.
Good luck with whatever you decide is best for you.