i would not recommend traveling on a car, the road police a lot of times is acting unlawfully and will frame you up
please do not try to bribe anyone, even if conditions allow to
be particularly wary of extortion at the customs, it can come in many guises
if medical workers demand money for something covered by your medical insurance don't give in, unless of course your health profoundly depends on your decision.
keep a mobile phone to use for calls within the country you're in and a couple of numbers of the US consulate in that country
do not hesitate to threat will a phone call to the consulate if any official is harassing you
if you happen to have to protect your rights, basically do it as you would do so in the US
in Ukraine police emergency number is 102, ambulance 103
don't try to bring into Ukraine gadgets with built in video camera (except cell phone) this is prosecutable as criminal offence
of course it's best if you have someone from the locals keeping company with you