The nearest significant town on the Polish side is Przemyśl, and it's easy to find by following route #4 (which passes through Przemyśl), also known as the E40 in European terms.
When you arrive, the road is fairly narrow (no motorway/autobahn this) with a queue of trucks and vans parked to the right of the road; a hard-core parking area with cafe/bar to the left. Don't stop 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 - commercial traffic goes through a different process).
If you're in an EU registered car then make for the EU-passports, passport control section. Thence to Ukrainian passport control and then Ukrainian customs and then you're through. It used to be a nightmare, with apocalyptic tales of 5-6+ hours at the border, but the Ukrainians have made great advances in efficiency and it takes about an hour to make the crossing (2012). Don't expect the border police to treat you in a friendly or even respectful manner, in fact, expect anything ranging from neutral to extremely obnoxious behavior.
Once through, just follow the main road towards Lviv on the E40 - this is the route right across Ukraine to Kiev (and thence on to the east). Stick to this - the main towns on the way are Lviv, Rivne, Zhytomyr.
Watch out about 15-20 km inside Ukraine, in Mostyska, as police have gone crazy about traffic calming measures here (speed bumps or sleeping policemen). They are like icebergs across the road, and very badly marked. There are about four or five sets of them through the village.
Other than that, take care on the road, which although the main east/west highway, and the main road route into the EU, still remains in a miserable condition (surface-wise). You will soon realise why Ukraine has such poor statistics in relation to driver and pedestrian