This is an excellent question, because the time period you chose is one of major technological innovations. The ship types developed in this period essentially dominated naval transport and warfare in the Mediterranean for the remainder of Antiquity.

In the 7th century BC, as far as we can tell, just about all forms of naval activity were carried out using large galleys called pentekonters (literally "fifties"). The name comes from the fact that, in addition to a single rectangular sail, they were powered by a total of fifty rowers, arranged on a single level in two parallel rows. This ship type was developed from early Late Bronze Age oared vessels; it shared the characteristics of shallow draft and slightly upturned, pointed prow, which allowed for easy landfall on beaches.

Pentekonters were privately owned. Greek states did not yet have the degree of control and financial resources to build their own navies; at best, they could require wealthy ship owners to provide a ship and crew for the defence of the community in wartime. At all other times, the owner of a pentekonter could use it as he saw fit. The impression we get from Homer and Hesiod is that when the sailing season came around, ship owners would set out on a voyage that could involve raiding, piracy, trade, or simple social calls on overseas guest-friends - all depending on who they encountered and what seemed expedient in the circumstances. The pentekonter was the all-rounder that made this flexible form of income gathering possible. The ship was seaworthy and capable of long-distance journeys; its crew of fifty made for a sizeable force if raiding or piracy was on the menu; if trade was the goal, there was enough space in the hull at least to carry the luxury goods that made up the bulk of the trade goods of this period.

In the course of the 7th and 6th centuries, however, the political situation changed. As states gained more control and the role of elites became more institutionalised, the tendency to go off on a bit of private warfare or piracy came to be frowned upon. Increasing stability and prosperity generated an intensification of trade. At this point we see the first separation of ship roles. All forms of raiding and warfare continued to be the preserve of the many-oared, powerful pentekonter. But from the mid-6th century BC, we see the first depictions (and to the same time can be dated the earliest shipwrecks) of dedicated cargo ships.

These ships were totally different from the sleek pentekonter. They were tall, with wide bodies, and were propelled only by a single tall square-rigged sail. They were slow and defenceless, but they could carry vastly more cargo than the pentekonter, which allowed bulk trade in things like grain, wine or oil to become profitable. Their arrival marks a shift from trade as a way for the rich to get a bit more money on the side to a legitimate livelihood for dedicated merchants. This ship type remained the standard trade vessel throughout antiquity. The type was upscaled gradually to the point where Roman grain ships could carry as much as 450 tons of cargo.

The next major change came in the late 6th century BC. As states began to construct navies and naval warfare became a more large-scale affair, the realisation came that larger ships with more rowers could be faster and more manoeuvreable than the stalwart pentekonter. We don't know exactly when the bireme was developed from the pentekonter, adding a deck of rowers and a ram; certainly Polykrates of Samos' powerful navy of the 520s was still equipped with pentekonters (which in any case remained in service as lesser warships for centuries afterwards). But from that time onwards, Greek naval warfare came to be dominated by an innovation from the east - the trireme.

In its design, the trireme was mostly an upscaling of the pentekonter's principles. It added a third bank of oars and was larger, and with trained crews its ram became a primary weapon, but otherwise it functioned in mostly the same way. However, there was one major difference: the trireme was exclusively a warship. Its crew of 200 left no space for cargo; its dependence on speed and balance made it impossible to store anything on deck. It still carried a sail for long journeys, but when battle was imminent, it was stripped of its mast and rigging, minimising weight and allowing the rowers to turn the whole ship into a battering ram.

Despite its apparently singular purpose, there were actually two major ways to use the trireme in battle. One of these our sources present as old-fashioned and crude was to board enemy ships and use deck-fighters to fight what was essentially a land battle on deck. The other, which our sources consider the prerogative of experienced crews, was to use the ship ram to damage the hulls of enemy triremes, capsizing or sinking them. This required a lot more control and speed, requiring the whole crew to work as a well-oiled machine. Unsurprisingly, there were few skilled workers as desired on the job market as experienced trireme helmsmen.

Wars between Greek naval powers in the 5th century BC spurred several innovations. Athenian triremes were initially without a full deck, but the later "Kimonian" triremes had a full deck, protecting the rowers from missiles and allowing more flexibility for the ship crew. In the 420s BC, the Athenians first introduced horse transports for their overseas campaigns, which may have been based on converted merchant ships.

Yet the trireme remained the basic warship of all Greek naval powers until the Greeks of Sicily got involved in a new war with Carthage in the early 4th century BC. This spurred a new innovation: adding more decks of rowers. Dionysios of Syracuse's engineers are credited with inventing the quadrireme and the quinqereme (as well as the catapult) around 399 BC, and much of Syracuse's navy was equipped with these "fours" and "fives" in the ensuing war. The new larger ships took some time to influence naval warfare further east, but by the end of the fourth century BC, the Athenian navy boasted many quadriremes and quinqeremes alongside its trusty triremes. In the Hellenistic period, upping the ante with regard to oars led to an arms race between major naval powers like Syracuse, the Seleukids and Ptolemaic Egypt, and the development of sixes, sevens, and sixteens, but that is outside of the scope of your question (thankfully, because I know very little about it!).

Follow-up:

Is it accurate to refer to biremes, triremes, and the like as galleys or is a galley a later kid of ship that evolved from those?