By car
The main route into Ukraine from the West is via Poland - the only 24-hour customs post is in Lvivska Oblast Region at a place called Krakovets. The nearest significant town on the Polish side is PrzemyÅl, and it's straightforward to find by following route #4 which passes through PrzemyÅl. When you arrive at the border, the road is fairly narrow no motorway/autobahn, and there is always a queue of trucks and vans parked to the right of the road. Don't park behind the goods vehicles, slip up the side of them and then feed into the customs area when the guy flags you forward for courteous Europeans, you're not jumping the queue as commercial traffic goes through a different process. If you're in an EU-registered car then find the EU-passports section. Then, proceed to Ukrainian passport control and then Ukrainian customs and you're through. It used to be a nightmare, with apocryphal tales of 5-6+ hours at the border, but the Ukrainians have made great advances in efficiency with a 1-2 hour border crossing now possible.
Once through, just follow the main road towards Lviv ÐÑвÑв on the E40 - this is the route right across Ukraine to Kiev and thence the East. Stick to this - the main towns on the way are Lviv, Rivne Ð Ñвне, Zhytomyr ÐиÑомиÑ. Care is required as the road still remains in a miserable condition, even though it is the main East/West highway and the main road route from and to the EU.
By ship
It is possible to organize trips down the Dnieper to the Black Sea in summer. A travel agency in Ukraine can book these trips for you.