The beach choices close to the city aren't the nicest. Just west of the city (Bus 6 from Hotel Astoria in Plateia Eleftherias goes to the western beaches, and from the stop opposite Bus 7). Ammoudari (also called Almyros) is to the west, but the surroundings include cement and power plants and big hotels.
To the east of the city 7 km is Krateros (Bus 1, also from the Hotel Astoria), and--an improvement compared with the others is Amnisos, which is the first of the string of resorts to the east of Heraklion. But the latter two are close to the airport, so no quiet idyll is possible if you don't travel farther east. Amnisos has several tavernas, and sits up over an islet (Dia-for Dias, or Zeus) which is a sanctuary for the kri-kri, an endangered ibex indigenous to Crete. It was a port of Knossos, and from here ships sailed to Troy, and Odysseus' ship was stranded by the north wind. The rightly famous Fresco of the Lilies found in the Heraklion archaeological museum, was found in the 1600 BC villa on a hill above the beach. There were two harbors here, and while excavating the Minoan Harbor-master's office during the 1930s, Spyridon Marinatos found the layer of pumice that gave support to his theory that Minoan civilization had been destroyed by the ash-fall from the explosion of the Santorini volcano.
Knossos and Phaistos along with the Heraklion Archeology Museum are the three must see major Archeological sites of Crete. Phaistos you'll read about below but Knossos and the Heraklion Archeology Museum each receive its own treatment.