Eastern European cities generally have good public transport.
The Schengen Agreement, which allows for passport-free travel within a large chunk of Europe, includes the Czech Republic, Estonia, Hungary, Latvia, Lithuania, Poland, Slovakia and Slovenia. EU member countries that are not part of the agreement – and where the usual border stops and passport checks apply – include (at the time of research) Bulgaria, Croatia and Romania. For up-to-date details see www.schengenvisainfo.com.
A somewhat relaxed version of border control has been reestablished between Russia and Belarus, despite the two being part of a single customs union. You must have visas for both countries; crossing without them is a criminal offence. Don't even consider entering Belarus on a Russian visa or vice versa.
There are potentially serious implications for those transiting into Russia via Belarus on an international bus or train as you will not receive a Russian border stamp or an immigration form on entering the country. If you later plan to exit Russia via a different route this will be a problem and you could be fined.
We’ve not heard of any travellers running into serious difficulties but it would still be wise to make careful enquiries with visa authorities in both Belarus and Russia before you’ve confirmed your travel arrangements.
The two countries were essentially at war with each other at the time of writing, but it was still possible to cross in both directions by vehicle or train, with the exception of rebel-held zones in southeastern Ukraine and the Russian-controlled Crimea. Crossing into the rebel-held zones or Crimea from the Russian side is a criminal offence under Ukrainian law. Ukraine has also announced that it will no longer permit foreigners to travel to Crimea from the Ukrainian side.