When Cyprus achieved its independence from Britain in 1960, the new Republic’s constitution defined the Greek and Turkish Cypriots as two separate ethnic groups.
At that time, the members of both groups still co-habited in mixed villages and cities and purely Greek or Turkish villages were few. In the majority of cases, neighbours lived together in peace and celebrated their festivals together.
Then the events of 1974 split the island, and the two communities have not lived with each other for over 30 years.
Nevertheless, the two communities still have a lot of things in common: in the way they conduct their lives, their gestures, their food and drink. The Greek-Cypriots and Turkish-Cypriots on both sides of the Green Line are all Cypriots, after all.
The official languages of Cyprus are Greek, Turkish and English. Visitors have no trouble communicating in English due to the fact that Cyprus was a British colony from 1878 until 1960, and even today the British maintain military bases on the island.
While you can get by this way, the ability and willingness to say even a few words in Greek will definitely be warmly welcomed and may "upgrade" your status from a simple touristas (or tourist) to a more honourable xenos, which means foreigner or traveller.
It is important to note that the Greek spoken in Cyprus is a strong dialect, with about 15% of the words peculiar to Cyprus. This means that people from Greece often find it hard to understand a conversation among Greek Cypriots.
Greek is not an easy language. The grammar is complicated with nouns divided into three genders, all with different ending cases in the singular and plural, and adjectives that have to agree with these in gender, number and case.
Then comes the verbs which are even more complex; they come in passive and active voices, in two conjugations, and so on. But this should not be a discouragement to learn at least the basics.
If you are interested in learning Greek, other than taking lessons at an accredited institute, you can chose the option of teaching yourself with the aid of a book or online course.