The answer really depends on where and when specifically you're asking about. You're right that triremes aren't very versatile ships. They're too narrow for cargo or passengers, they hardly even accommodate the crew. Triremes would have to come ashore every night for the crew to camp. Sailing and warfare are seasonal and for the rest of the year the ships have to be kept in shipyards and carefully maintained.

Early naval history is not super clear. It seems likely that early navies were privately maintained by aristocrats and consequently highly idiosyncratic. Athens (which had already started shifting to a city owned fleet) greatly expanded their navy after finding a new seam of silver at Laurium in 483, ostensibly to go to war with the Aeginetans who were both a dominant naval power and uncomfortably close to Athens. Afterwards, the fleet was maintained by liturgies paid by the wealthiest citizens (not by direct imperial tribute). While we don't have fantastic sources on how the system worked in other cities in the early period, it is likely to be a similar system of taxes on the wealthiest citizens or privately owned and maintained navies by aristocratic families. We aren't talking about a modern fleet here. At the height of their supremacy, Athens had a fleet of 200 ships. It's unlikely that smaller city-states had anywhere near that number.

In Athens, these trierarchs who provided the liturgies were expected to keep the ship maintained, finance new ships, and supply crews. There's a lot of debate still on the exact makeup and training of these crews. Warfare and sailing are both seasonal labor, it seems that a lot of the crewmembers were thetes - the lowest social class who did not own land and therefore had no stable income. They were paid wages for their service, but not enough to sustain them year round, so they would have had to find other forms of paid labor when they were not serving. It's likely that they helped with farming, it's also likely that most craftsmen belonged to this class. They also received a certain amount of public support. Athens prided itself on its navy of free citizens, but it is likely that there were resident foreigners and some slaves contracted out as sailors and that in desperate times (as in the Peloponnesian War) slaves served in the Athenian navy. Other cities appear to have had slaves serve in their navies, but our sources tend to prefer to ignore slavery, making evidence a little thin on the ground.

Outside of Athens, it's hard to be sure. Island and Ionian states (like Lesbos, Chios, Miletus, Samos, etc) had fleets of triremes and penteconters for fighting, but took heavy losses during the Persian Wars that they never quite seemed to fully recover from. During the height of their supremacy Athens did it's best to hamstring the naval power of their allies. While the early terms of the Delian League allowed for cities to contribute ships under their control to the navy in lieu of monetary contributions, their allies navies eventually deteriorated allowing only an Athenian fleet. On the Peloponnesian side, the Corinthians had the largest navy with something like 90 ships. As far as I know (though I work on Athens, so I could be wrong), we have almost no sources - historical, archaeological, or otherwise - that tell us anything specific about the financing or crews of these ships. When hostilities broke out with Athens, the Spartans turned to the Persians to finance their new fleet. Eventually, after their heavy losses in Sicily the Athenians turned to any possible sources and gave away the last of their reserves to try to rebuild their navy, but combined with the poor decision to execute some of their most experienced commanders, it wasn't enough to recover.