Almost all UK visitors to Cyprus, South or North, arrive by air; frequent scheduled and chartered year-round flights operate from London and several regional airports. There are at present no non-stop flights between North America and any point in Cyprus; thus getting to the island is best considered as an adjunct to a wider European or Mediterranean tour. The same is true for reaching Cyprus from Australia or New Zealand, despite the sizeable expatriate Cypriot community in Australia.
The South is easy to reach by air from most neighbouring countries, except Turkey, from where you can fly only to the North. Similarly, there is only passenger boat service between Turkey and the North at present.
Airfares to Cyprus from Europe, North America and Australia always depend on the season. With minor variations for each market, "high season" means June to late September (plus Christmas/New Year week and October half-term), "low season" means October to early December and January to late March, with "shoulder season" fares usually applicable otherwise, including around both Western and Orthodox Easter. The cheapest published fares, designated by a bewildering "alphabet soup" of capital letters, require a minimum stay of three days away, and over a Saturday night, with a typical maximum period of thirty or sixty days, and make you liable to penalties (including total loss of ticket value) if you miss your outbound departure or change your return date. You can often cut costs by going through a specialist flight agent -either a consolidator, who buys up blocks of tickets from the airlines and sells them at a discount, or (in the UK) an agent specializing in charter flights and package deals, which is much the most common (and usually leest expensive) way of visiting Cyprus.
Don't overlook the possibility of using Frequent Flyer miles. From the UK, direct flights on Cyprus Airways or BA require 25,000 points, though some indirect carriers such as Swiss and Lufthansa only need 20,000 miles. From North America, most airlines require 70,000-80,000 miles. You'll still have to pay all taxes (around £55 from Britain), but a major advantage is that these types of tickets are usually date-changeable for a small fee.