It's easy to learn, and absolutely worth it, even if you don't know any thing else in the languages. It helps most with signs, subways, maps and directions and it will also just raise your general comfort level--especially in Ukraine, Belarus, Bulgaria, Makedonia.
You should also learn the latin alphabet of Former-Yugo. Same sounds as Cyrillic, just with latin letters--and only a handful of extra characters than English. You will probably use that alphabet more if you don't go past Serbia.
Be careful about which sites you use, as they're not all completely accurate with pronunciation. Some are even missing letters for the respective language. Since Cyrillic spans multiple languages, you may want to start with one language and learn some basic phrases to give the script more context. Russian is probably your best bet as there are many resources available and they will always be in Cyrillic (as opposed to Serbian, which would be my second choice since you'll be in that area).
Technically, it's a little confusing to just learn the Cryillic script without focusing on one language, because although they share most of the same characters, there are some unique ones to each language, and even some of the shared characters have slightly different pronunciation.
But don't let this confuse you. Just pick a language with Cyrillic (Russian!). And then, once you know the alphabet and a few words, switch to another Cyrillic language and learn the small differences and extra (or substitute) characters. There may be a site that lists all characters (like wiki), but again, I think this is too confusing and not at all a natural way to learn.
When I have some time, I'll look for more sites, but here is one for now.
http://www.friends-partners.org/oldfriends/language/russian-alphabet.htmlAgain it helps if you have context and can
hear a few words being pronounced.