The locals have recently decided to officially name the language they speak Montenegrin. This is a dialect of Serbian that has been promoted as the national language since independence in 2006, though many locals still declare they simply speak Serbian. Whatever you want to call it, it's a south Slavic language closely linked to others in the region, but with small variations and an alphabet that has two extra letters.
As in Serbia, both the Latin and Cyrillic can be used to write Montenegrin, though nearly all texts, street names and newspapers are in Latin script. While we're on the topic of language, note that people from Montenegro are not Montenegroes as you sometimes see written, but Montenegrins.

Pronunciation
Č č    like ch in church
Ć ć    similar to č, but softer
DŽ dž    like g in George
Đ/DJ đ/dj    like j in jaw
J j    like y in yes
Š š    like sh in show
U u    like oo in roof
Ž ž    like x in luxurious

Niceties & Necessities
Yes           Da
No            Ne
Hello          Zdravo
Goodbye          Do viđenja
Good          Dobro
Please          Molim
Thank you        Hvala
Here you are      Izvolite
Good morning      Dobro jutro
Good day        Dobar dan
Good evening        Dobar veče
Good night        Laku noć
Cheers!         Živjeli!
All the best!       Sve najbolje!
Excuse me      Oprostite
Sorry        Izvinite
How are you?      Kako si?

Practicalities
My name is...        Moje ime je…
I’m from...          Jas am iz…
...UK            ...Velika Britanija
...USA            ...Amerika
I don’t understand      Ne razumijem
I don’t speak...         Ne govorim...
... Montenegrin/Serbian     Crnogorski/Srpski
What’s your name?      Kako se ti zoveš?/Kako ti je ime?
A ticket, please       Kartu, molim
How much does this cost?  Koliko ovo košta?
When?            Kada?
Now            Sada / Odmah
Where?           Gdje?
Here            Ovdje
There            Tamo
Who?            Ko?
Why?            Zašto?
What?            Šta?
I have...           Ja imam…
I am...            Jas am…

Signs
Open              Otvoreno
Closed             Zatvoreno
Entrance          Ulaz
Exit             Izlaz
Push              Guraj
Pull              Vuci
No smoking        Zabranjeno pušenje

Numbers
0 - nula      
1 - jedan       
2 - dva
3 - tri       
4 - četiri       
5 - pet
6 - šest      
7 - sedam       
8 - osam
9 - devet     
10 - deset       
20 - dvadeset
50 - pedeset   
100 - sto       
1000 - hiljadu

Days
Monday          Ponedjeljak
Tuesday          Utorak
Wednesday         Srijeda
Thursday          Četvrtak
Friday            Petak
Saturday          Subota
Sunday           Nedjelja

Months
January          Januar
February          Februar
March            Mart
April            April
May            Maj
June            Jun
July             Jul
August            Avgust
September         Septembar
October          Oktobar
November          Novembar
December          Decembar

Time
Now            Sada
Later            Kasnije
Today            Danas
Tomorrow          Sjutra
Yesterday          Juče
In the morning        Ujutru
In the afternoon      Po podne
In the evening  Uveče
At night   Noću