Where (what city) do you want to travel? It is possible to rent a car but it is sometimes easier to rent a driver with the car. By the way how much is the car rent in your country? And of course you have to get used to local road police, awful roads and impolite drivers.
I am traveling to Odessa. It will be my second time in Ukraine. My First time was 8 days in Kiev and Donetsk. What about buying a cheap car? Is insurance an issue?
http://www.hertz.com/ most people rent their cars there, in Odessa they are situated at the Sea Port building, you can rent a car online. Nothing good in buying a car in Ukraine
An alternative is to hire a car and driver.
I wouldn't fancy driving around on my own in Ukraine.
A good English speaking driver can double as a guide.
I would do this if I ever went to Ukraine again.
Make a deal with the one of the taxi drivers at the airport if possible.
Do the guy a decent deal, dont try rip him off, you want him as a friend.
That's what I'd do.
I was with a girl in Simpferopol once. She got the bus to meet me at the airport and we took a taxi into town.
I took the first guy who approached me... just said..."ok" thumbs up gave him the international sign for ok...thumb and forefinger hoping it wasn't Ukrainian for something else.
He was a bit confused at first when I stood there scanning for the girl, but got the picture when she appeared.
For the next two weeks this guy (Mickelo) was at at our beck and call.
She just called him and he came.
I don't even think he was a taxi driver...he just had a car...Top of the range Lada (I think...and possibly 15 years old). Electric rear windows and a set of warning lights on the dash in various colours which stayed lit the whole time.
But that car took on every pothole and seriously dodgy road from Simpferopol to Yalta to Sevastopol and back again.