Paul has given you good background information.
I passed through Serbia twice rather briefly in 2015, spending 2 nights in Beograd and one in Nis. At the time I observed that the Latin script had made major inroads. Some signs on shops seemed to have no Cyrillic at all. Still, I think it will be prudent to learn the Cyrillic alphabet so you can sound out things like geographical names if you need to. (There's some bleed-over to Greek as well.) Looking at the names of major cities may be helpful in memorization.
As another former student of Russian, I prefer dealing with Cyrillic because to me it fits better with the Slavic sounds of the language, but if your time is very limited, I think you'll manage pretty well if you spend a bit of time studying words in the Latin form. I'm heading to Hungary this year, and--though Hungarian theoretically uses our alphabet--I fear that they'll be lucky to hear "Good morning" and "please" from me in some semblance of their language, which is one of the screwiest things I've ever seen (no offence meant to any Hungarians who may read this post).
The good news is that Serbian and Croatian are essentially the same language, just historically written in different alphabets, or so I was told back in the 1970s. I think the same is true of what are now called "Bosnian" and "Macedonian". Slovenian is a totally different animal, though Slavic at its core; I'm not sure I understood a word. At a high level, I'd say Slovenian is to Serbo-Croatian (yes, there used to be a common name for the two languages, at least while Tito was in power) as Romanian is to Italian--not as close as a traveler might wish.
In a pinch, look for a young person to help you out in Serbia. Many will have studied English, though it was my impression that the language isn't as widely spoken as it is in Slovenia and Croatia, which simply seem to get a lot more western European and North American travelers.