Since the take over of the Crimea by Russia, and the ensuing violence in the east of the Ukraine, it’s a delicate question as to what language you should be speaking in your attempts to woo a Ukrainian girl. Most Ukrainians are bilingual, speaking both Ukrainian and Russian, but whether they prefer to speak one or the other depends on geography. In Kiev, you will definitely find Russian more useful. Outside of Kiev, broadly speaking western Ukraine speaks predominantly Ukrainian, and eastern Ukraine is more linguistically Russian.
In some western cities, although Russian is understood as good as Ukrainian, you will find that since the troubles began, most people will prefer to speak in Ukrainian, and may even be offended if you, as a foreigner, speak in Russian. Lviv in particular is known as a very nationalistic city.
Both Ukrainian and Russian are slavic languages, and both use the Cyrillic alphabet. However, the grammer and the sound of Ukrainian is closer to Polish than Russian.
Here is a useful YouTube video from the beautiful Ukrainian girl Taya Ukraine, talking about which language is more important to learn for foreign men seeking love in her country.