The economics of a Star Trek based universe are more than slightly unbelievable. While the most recent references made are that society no longer uses money and the acquisition of wealthy is no longer the driving force in lives of most of those within the Federation, there are numerous on-screen references to currencies of some kind. This makes the economics of the universe a bit uncertain.
Energy Economy
Since such things as the matter/antimatter reactor and cold fusion reactors exist coupled with the presence of replicator technology it is safe to say that the economy of most worlds runs on some form of energy. Energy can be used to produce nearly any goods that an economy might need to operate, from building materials to living materials. Since modern societies operate with an incredibly high capacity to produce energy (matter/antimatter in particular produces a nearly totally mass-to-energy conversion), that leaves most societies in a state of energy surplus, which accounts for the very high standards of living without the focus on acquisition of wealth in most societies.
Fuel
Since the primary source for energy within this kind of economy is not solar, which would of course be free of a fuel source in this context, the most precious commodity in an energy economy is fuel. In the case of the Star Trek universe, the most common fuels are deuterium and anti-deuterium. Fusion reactors run on straight up deuterium, which is a specialized form of hydrogen which has one proton and one neutron in the nucleus rather than just one proton. This allows for the deuterium to be fused together to form helium.
Within a matter/antimatter reactor deuterium is collided with anti-deuterium (it's exact opposite in terms of subatomic particle charge). This is the most efficient means of taking matter and extracting energy from that matter there is for most Federation worlds. It should be noted here, however, that some worlds (such as the Romulan Star Empire) use other methods for converting mass directly into energy, in the case of the Romulan's they use an artificial quantum singularity which converts its mass into high energy plasma.
Replicating Fuel
One might easily conclude that a person could replicate fuel, since most things can be replicated. The problem is that the laws of conservation of energy and matter along with the 2nd law of thermodynamics combine to ensure there is a net loss when creating matter from energy and when converting energy to matter. The process itself, in either case, costs energy. Thus replicating fuel is possible but not cost effective.
In the case of deuterium (hydrogen being the most common element in known space), it is far more effective to gather up hydrogen from space or from world with a high concentration of the substance (such as Jupiter in the Terran system) or worlds that are predominantly water. Antimatter, on the other hand, cannot be replicated safely and must be created from deuterium using subspace manipulation.
Antimatter
Since antimatter is neither abundant nor naturally occurring, according to the Star Trek the Next Generation Technical Manual, antimatter is produced through a vague process involving "charge reversal" devices, resulting in a 24% loss of energy for the process. This makes antimatter one of the most, or the most, precious commodity of fuel for energy generation in known space for races which use it. There are no canonical details on how Romulans create an artificial singularity, so one cannot speculate on what they might consider the most precious fuel commodity.
Since antimatter is so difficult to make, it is mostly produced by planetary / system wide government or corporate enterprises. Operations of that size maintain the resources to engage in such expensive operations and would grain significant authority for providing fuel distribution services. As such, Warp Drive and the matter/antimatter reactor would be subject to legal provisions, government sources would undoubtably require some kind of licensing for distributors, and in turn those would require valid space-fareing licenses and registration for vessels as not to be potentially implicated as accomplices in crimes.
Dilithium
The reaction between matter and antimatter is regulated through dilithium crystals. Without these crystals the reaction cannot take place in a safe way, thus throughout canonical Star Trek and its various series, movies, and the like dilithium crystals are looked upon as a fuel source. They do get used up, as the reaction breaks down the crystal latices (as explained in Star Trek IV the Voyage Home), Scotty refers to this as them being "drained", which goes along with their place alongside fuel sources. They are a consumable fuel commodity, without which interstellar travel cannot take place using Warp Drive (except in the case of Romulans, though this is not canonical since it is contained in a series of Enterprise novels which are not part of canon, the show does not ever say if dilithium crystals are used by the Romulans).
It is fairly well understood throughout the shows that if they could replicate Dilithium, they would and multiple times throughout the shows, even those of DS9 and Voyager, that at times they are in dire need of the substance and need to seek natural sources in which to find it.
Currency Economies
Replicator technologies combined with computer advancements make most currency systems obsolete. Money in the forms of coins and bills can replicated with relative ease. To enforce a system of currency requires scarcity of whatever the currency is made of. Gold is no longer usable as it can be replicated, similar for nearly any other substance. In the Star Trek universe, the Ferengi are the most prominent users of currency, as we know it today. Paper currency notes (dollars, pounds, yen) have no real value to them other than what is ascribed to them by the fact that they are so difficult (and illegal) to produce that they then are given value through our exchange of them for goods and services.
Similarly the Ferengi and other races use a currency known as Gold Pressed Latinum. In this, the gold is infused with a substance that is impossible to replicate known as Latinum. The gold itself has value, of course, as it is a heavy element that is difficult to procure naturally, but can be replicated. The mix of the two, however, create an item that is both valuable, easily transported, and cannot be replicated. Within that canon of Star Trek it never says where Latinum comes from. In several games, it comes from a specific kind of nebula where as in some of the non-canon novels it comes from crystals that can be mined. In either case, the Federation has little use for the substance other than trade with races who use it and scientific research.
Economic System of the Federation
While most people in the United States would not taking kindly to this idea, it appears that the United Federation of Planets is some form of socialism, able to work with established democratic freedoms because of the energy abundance. In many episodes, especially those surrounding Star Trek Deep Space 9, it is seen where the Federation government officials take the collectivist view against even large numbers of colonists for the advantage of the state. Now, it is also clearly stated that the United Federation of Planets is a representative republic where free elections are held on all member worlds to elect senators who attend the Federation Council.
While planets must have governments which share Federation values, it does not anywhere state in canon that they must be democratic in nature, though it is heavily implied. The TV shows and movies depict a world similar to that of the United States of America, they are have "done away with petty differences" and of course being an American TV show it is very centered on American values. However, if one takes a longer view, the economy does not run off of exchange of wealth (capitalism) but rather runs off of the free and open cooperation of members of society to better themselves and each other. This collectivist view is far more similar to socialism as an economic system than of capitalism practiced in the United States.
Thus it can be figured that resources are divided based on the energy required to produce them through some method taking into account need of the individual (it says there is no poverty multiple times) as well as the need for / usefulness of the person in terms of productivity and work done. This is never said directly, however people eat, have some method of acquiring items, giving items in trade, and all of the things that go along with living ones life - all without the ideals of capitalism.
Economics of Space Travel
A varied economy would have multiple means of interstellar travel for different level of income. An efficient system, even one not capitalistic in nature, would not wish to use a resource intensive method when another one is available. While in Star Trek canon, warp drive is the only method available for interstellar travel, in Solas Tempus there are a variety of methods.
Warp Drive
The most expensive form of travel through space is that of Warp Drive. It requires the use of a matter/antimatter reactor, which operates using two of the most valuable commodities (Dilithium and Antimatter). Thus, it is understandable that an interstellar economy would have use of methods of travel which are not so costly. Governments would find it easy to produce vessels capable of warp travel and supporting the infrastructure to fuel such vessels. Similarly large corporations could do the same thing, provide infrastructure for the support of interstellar travel with the freedom involved in travel using Warp Drive.
Within the story, the derivative of Warp Drive, the Dimensional Fold Drive, is even more expensive to build and operate than the conventional Warp Drive system.
Hyperspace Travel
On the other end of the spectrum is travel through the Jump Gate network, which within the storyline exist at every major outpost and planet in the galaxy. These are somewhat slower than warp travel but are reliable and operated by (mostly) governments and use of them is included in any licenses acquired to operate space crafts / space craft registrations. These are the equivalent of the trucking highways, railways, and waterways traveled by cargo vessels. It is safe to say that, by and large, most cargo that doesn't need to be there quickly runs through hyperspace within the Blazing Umbra Setting.
Gravitational Displacement Drive
For smaller vessels (scout ships, interceptors, shuttles, etc...) the Gravitational Displacement Drive allows for small vessels (and medium sized ones, though at a premium on cost in energy) to instantaneously travel from one point to another provided they can start at a sufficient gravity well. This also means that there is a support infrastructure for these vessels as well, including designated jump areas to depart from (near large gravitational bodies such as stars and similar stellar phenomena) and arrival at (cordoned off areas that are kept clear in case of unexpected arrivals).
Vessels that have both a warp drive system as well as a gravitational displacement drive system as the same integrated unit (using a matter/antimatter reactor to produce lygon particles which then warp gravity) would give up the economics of using the drive system in favor of having the instant gratification of jumping from one location to the other. Even a small vessel would need to use the Oscillating Mercury Core as a power source to fit within the economics of using such a drive system. Though giving more freedom to operators of such vessels (instant jumps as well as warp drive), smaller vessels would essentially be relegated to going from star to star in order to maintain a stellar body nearby to jump from.
Since such vessels would not require the use of expensive warp drive nor require the use of jump gates, the GDD is popular among less than "mainstream" operations, such as smugglers and criminal enterprises. Many of these vessels have two separate engine cores, one for Warp Drive and one for the GDD. Such a configuration would be costly on space but allow for economic flexibility. As discussed before, the sale of antimatter would mostly trace back to governments or companies which operate under government contracts and thus be subject to legal ties. Similar with the jump gate network and use of hyperspace, which can only really be operated by governments or excessively large companies which would too be subject to government and legal ties.
Vessels with a GDD would thus be more heavily scrutinized at docking facilities and fueling stations because of the link between such a drive system and criminal activity, but the drive system would allow more freedom to operate under the radar of law enforcement.
Stargates
The recently discovered Stargates also now exist at most planetary hubs and large populations. As a means of conveying people and small quantities of goods, they are superior to most methods of space travel. They are included here only because they have the ability to supersede other forms of space travel through the ease with which they operate. Since there are a limited number of gates, by and large the only gates accessibly are operated by government entities and thus require the normal trappings of legitimacy required for using government facilities.