Lviv
Lviv is a large city in the west of Ukraine and the unofficial capital of Galicia. For a traveler through the Carpathians, it often becomes a gateway to the mountains: they fly here, arrive by train and from here go south to the ridges.
The city itself is well worth a stop for a day or two. The old town center with Rynok Square is included in the UNESCO World Heritage List, and it is easy to lose half a day among the cafes, churches and cobblestones, even if you only planned a transfer.
So if the route leads to the Carpathians, Lviv is a logical starting point: here you can sleep and have a humane breakfast before the road to the mountains.