Though official language of Ukraine is Ukrainian, more than 60% of its population speak Russian, because of long lasted and close bonds between Russian and Ukrainian people. For more than 300 years Ukraine was a part of Russia, and lately USSR. Last generations of Ukrainian people generally learned in school and spoke Russian in everyday life.
In 1991 Ukraine one more time became independent. Ukraine is a “young country”, which is trying to reestablish its ancient historical roots. Different provinces of Ukraine have their own traditionally formed ethnic compositions and corresponding preferences, and it reflects in prevalent speaking language.
Thus, in the Eastern part of Ukraine with close historic bonds with Russia people mostly speak Russian (Kiev, Dnepropetrovsk, Donetsk, Kharkov, Odessa, the Crimea Region). On a contrary, in the West of Ukraine with traditionally strong influence of Poland people mostly speak in Ukrainian language (Lviv, Ivano-Frankovsk, Rovno, Ternopol, Vinnitsa).
As Slavic people, Ukrainians are very similar to Polish, Russian and Belarusian people.
Most people understand English because English is a dominant mandatory foreign language in schools. You won’t have a problem in everyday communication in Ukraine, though you want to have a competent interpreter next to you in official and otherwise important situations.
You should try to speak in simple Russian or Ukrainian language. Ukrainian people will understand you. Just keep in mind, sometimes your pronunciation will sound funny to them, and they might smile. Don’t be discouraged by that. Ukrainians are very hospitable and heart-opened people. They will do their best to make you feel comfortable in their country.