All buses that travel along the GC500 coast road between Maspalomas and Meloneras and Puerto Rico stop at Arguineguín town. Most stop at the main bus stop along the GC500 (a two-minute walk from the sea) but some go into the town itself. The two main stops are the one in front of the La Marea shopping centre and the one by the Whale Tail roundabout (you can't miss it).
The most convenient bus is Line One (blue Global bus) as it travels the whole GC500 road and stops at lots of the quiet beaches between Arguinegiín and Puerto de Mogán.
Be aware that the road is closed between Taurito and Playa de Mogan due to a big rockfall so all buses currently use the motorway for this stretch.
From Playa del Inglés, catch Line One, or Line 33 which starts at the Parque Tropical bus station right at the east end of PDI. It stops in lots of places in PDI and Maspalomas before heading to Arguineguín and then on along the coast road through Patalavaca, Puerto Rico, Amadores and on to Puerto de Mogán.
Line 39 does a similar route but stops at Playa del Cura rather than continuing on to Puerto de Mogán.
You can also hop on Line 70 which runs from Puerto de Mogán to Palmitos Park via the bus station at Meloneras and Playa del Inglés resort.
A fast option is Line 91 which runs between Puerto de Mogán and Las Palmas via Puerto Rico, Arguineguín and Playa del Inglés. It's a fast option but doesn't stop at as many places as the other lines.
To get the bus to the airport from Arguineguín, or vice verse, take Line One or Line 91.