From the centre, walk to the massive blue-and-gold-roofed church just south. From here on, walk south on the Mynais'ka St. Continue on this street until you reach a good place to stand, just before the suburb of Mynai. The Mynais'ka St. will eventually turn into a big street towards Chop and the Chop-Záhony border crossing with Hungary.
North-West towards Slovakia
There are three ways of getting to Slovakia from around Uzhgorod: an ONLY walking border crossing in Mali Selmentsi, an ONLY train border crossing in Chop-Čierna, and an ONLY cars border crossing in Uzhgorod.
I. Pedestrian-ONLY border crossing in Mali Selmentsi is open till 20:00. There are buses (marshrutka) from Uzhgorod to Mali Selmentsi (and back) every hour for around 0.70 euros or you hitchhike there (there's not much traffic though). Only open to EU/EEA/CH and Ukrainian citizens.
II. Passenger train border crossing in Chop-Čierna n. Tisou. Trains go at 9:43 and at 17:35 from Chop and the ride lasts 1.5h and costs about 2 EUR.
III. Cars-only border crossing in Uzhgorod (the most complicated option): walk straight ahead from the train station main exit (westwards). After about 300m you find a big bus station. In the international service terminals (also called VIP waiting room : ) you can get a ticket to e.g. Vyšné Nemecké, which is just behind the border on the E50, for about 70 Hrivni (2,50 Euro), or Michalovce for about 250 Hrivni (9 Euro). Alternatively, you can take the public transportation marshrutka for 4 hrivnas: 19 (to KPP Masyv), 58 or 8 (to Era Shop) and walk ahead about 1 km till the border. BE AWARE: the border crossing in Uzhgorod is FOR CARS ONLY, therefore you will have to catch a ride before the border. This is not easy because even though most of the cars have Slovak number plates, they are actually Ukrainians who have to cross weekly only the Ukrainian border in order to keep the foreign registration of their cars. Plus, the queues there are usually looooong, so people are reluctant to take more passengers (more time consuming if the customs will check the baggage). But if you have time it is possible to cross the border there: a friend and I took 5 hours to cross it. ) If you want to cross the border faster then we advice to either take a bus, or to use the other two border crossing options.
Places to avoid
Don't hang around the train station at night, as the police will annoy you, ask for train tickets and passport. They also won't allow you to fall asleep on the chair. Best to go somewhere into a quiet street, or sit near the church just north-east of the centre.