Technically, Russian is commonly understood in any country that used to be part of the USSR: Ukraine, Belarus, Georgia, Armenia, Kazakhstan, Turkmenistan, Kyrgyzstan, Tajikistan, Uzbekistan, Azerbaijan, Moldova, Lithuania, Latvia, Estonia. This differs from place to place, and in more rural areas the use of their native language was much heavier than that of the official Russian, but generally people born as late as the 80's are bound to understand Russian.
Outside of the former USSR, Russian is understood by older people from countries that used to belong to the Soviet Bloc - Hungary, former Eastern Germany, Poland, the Czech Republic, and more so and even by some younger people in those where the the main local language and alphabet is similar to Russian, like Bulgaria, and where there is more trade and interaction with Russians today (near Kaliningrad, on the Black Sea coast, and in Prague and Karlovy Vary). There are parallel dynamics within the former non-aligned bloc of former Yugoslavia - Russian is most understand in Orthodox areas and on the Montenegrin coast.
However, in those places, as in some places within the former USSR, speaking Russian has political symbolism, depending on the exact context, that can be positive or negative.
There are also other favourite destinations of Russian-speaking tourists - places in Egypt, Tunisia, the Emirates and Turkey - where it is possible to visit using only Russian. In some neighbourhoods in Istanbul where many traders organise exports to the former Soviet Union, it is a lingua franca.
Unrelated to this, about a seventh of the population of Israel is made up of Russian-speaking immigrants and their children. You can visit Israel using only a working knowledge of Russian. There are significant Russian-speaking populations in New York, San Francisco, Seattle and many other places around the world, but the tourist infrastructure is not really oriented around Russian speakers in those places.