Montenegrins speak and write Montenegrin. It is sociolinguistically, ethnically, and culturally a separate language. For example, the Montenegrin language has 33 letters while Serbian and Croatian each have 30.
Difference between the languages spoken in Belgrade and Podgorica is that the latter has two extra letters. It is also characterized by its wide use of proverbs, metaphors and figurative speech. In Montenegro, according to the constitution, they officially speak the jekavian dialect of Serbian.
Most writing is in the Latina alphabet but Cyrillic is increasingly widely used, especially away from the coast and, unlike in Serbia, both alphabets have equal status under the constitution. Foreign tourists, especially those traveling around by car or bike, will find it helpful to have a basic familiarity with the Cyrillic alphabet, or at least a phrasebook with a crib.
Along the coast there are a number of minor dialects which are not too different from the mother tongue, as well as at least two varieties which are non-comprehensible to monolingual Serbians, one in Pastrovski spoken by the fiercely independent tribe who still predominate in the area around Petrovac, the other is Budva, spoken around that town. Neither is still widely used and intermarriage with outsiders poses a threat to their continuing viability. Native Albanian speakers are concentrated in the eastern border areas of country.
English replaced Russian as the second language in schools some years ago and there are generally people around in the resort areas with a reasonable command of the language. Elsewhere if you need an interpreter it is better bet to seek out a thirty-something younger. After Russian and English the most widely spoken foreign languages are Italian and German.