The airport of Recife was recently refurbished, and is today the largest airport in Brazilian northeast. The official name is Aeroporto Internacional Gilberto Freyre, but it's better known as Aeroporto dos Guararapes.
For tourists staying in Boa Viagem or in Porto de Galinhas, the airport is conveniently located.
Boa Viagem is a few minutes away; the distance from the airport main gate to the Boa Viagem beach is about 2 km. Notice, however, that depending on where a hotel in Boa Viagem is located, that distance could be up to about 8 km (but not much more than that).
The area around the airport is not very safe. Avoid walking around the area, especially at night, there is not much to be seen, anyway.
There are "special" taxis outside the airport; these are more comfortable - and safer - than the ordinary taxis across the street, but they are also more expensive (check for fares at the taxi balcony, to avoid rip offs). Also, there is a line of air conditioned micro buses (Line 42) with frequent departures from right outside the main gate; this bus travels all along Avenue Conselheiro Aguiar, a paralell to the main beach avenue, and can be a good option to save a few bucks, if you know where you are going.
From the airport, it's easy to reach the highway BR-101, which leads to Porto de Galinhas and the other beaches southern of Recife; there is no need to cross the city of Recife.
The inter-State bus station of Recife is called TIP - Terminal Interestadual de Passageiros. TIP is far from the Boa Viagem area and far from the downtown area; the terminal is on the road to the interior of Pernambuco, about 20 km away from Boa Viagem.
If you are coming from the South (Maceio, Aracaju or Salvador), try to get off as close as possible to the airport, and get a taxi from there. If you are coming from the North (Joao Pessoa, Natal, Fortaleza), get off at a stop named Caxanga (most passengers will know it; about half the passengers get off here).
If you have to go to TIP (advisable, depending on the time you arrive to Recife), there are a few options. Taxis are available, but a little expensive (check out fares with the taxi boots; they are open to some bargaining). There is a train station inside the TIP, but the trains don't go to Boa Viagem; if you go to Boa Viagem, take the train, get off at a station called Santa Luzia and take a taxi from there. There is a bus which goes to Boa Viagem, but it's best avoided, unless you have interest in a cheap two hour tour around half the city.