Advent Economy Tips?

A lot can be attributed to the fact that the Advent, as you've correctly noted, are not very economy focused. Their Trade Ships can't be upgraded and in order to use a Trade Port for Resource Focus, you need to sacrifice that Port's trade abilities.

To partly compensate for this, their ships, in relative terms, are rather cheap

You have to find a build sequence that optimizes the use of every single credit, and every unit of metal and crystal for about the first 10-20 minutes.  Even thereafter it will be very tight for most of the game.   There really is no means to save up for pirate bounty and keep up militarily or expand properly in advent, and if you play against better players you will almost always lose the game if you waste money on pirates at the start while playing advent.

Advent is about knowing what is important to build and what is not.   Without a tight control and streamlining of your budget you will lose.   That scarcity of resources is also why capturable asteroids are almost essential to get a slight edge playing advent.   Unfortunately, advent is about the worst civ to play to capture and hold capturable asteroids.   You could write an entire thread with the convoluted strategies for getting them as advent because it is so much more difficult and expensive.

Of course, another player could feed you, but then they would get behind, so it is better just to properly spend every single bit of resource in the opening minutes of the game on just the essentials of either illums if you are doing a very fast rush, or trade ports if you want to build a more robust econ first.   This advice is true for most maps because, unless you are on a huge multiplayer map, most maps will place you within 2 to 3 jumps from an enemy.   Most games therefore are rush or nearly rush games. 

If you build trade ports first, (going civic first), then it will take you 10 extra minutes before you produce your first illuminator.  If you go straight to illuminators your first illum will begin at the 10 min mark, and the 20 min mark if you go trade ports first.   That means you will only have disciples to help defend the first pirate attack if you go for trade ports first.   If you go illums first, defending the first wave of pirates is trivial since you will have illums coming out as they come in.   Therefore you also need to use the map chokepoints to help direct pirate attacks, and grab a dead asteroid or regular one nearest the pirate base.   Often, just by forcing pirate attacks on such a chokepoint, then fortifying the chokepoint with a few beam defense guns is all you need to survive a pirate attack... that allows you to continue expanding or even launching an attack against your enemy during a pirate attack.   Let your enemy waste his money on pirates.   You need to spend resources on structures, planet upgrades, or ships only.

Later in the game, if you streamlined and optimized at the start, resource and credit flow usually gets a little easier, but if you failed to do it at the start, you will almost always lose unless your opponent is a beginner.

Advent economy is the worst, unless the poor Vasari player can't find any neutrals...then he is in a lot of trouble too.

On the plus side, ton for ton, Advent ships bring the damage and have very good synergies once you get rolling.  They have the toughest scouts which can be used in an emergency to fight off LRM's or Assailants, they have the best carriers, and (arguably) best LRF, the Illuminator.  These ships are more expensive than their TEC and Vasari counterparts, but they are worth it.  Those drone hosts only take 2 labs, which is going to be pretty darn nice in 1.1.  Even the lowly Disciple has strength in numbers and more damage output than the other Basic Assaults for the cost.  Add in Iconus Guardians and get a couple capital ship synergies going, and you have a nasty force that can inflict a lot of damage. 

You could say that Advent's economy is in the deadly effectiveness of their fleets.  The enemy needs their economic advantages because they have higher losses, usually.

Advent military ships + caps means you wont be losing many ships.

Dont lose ships = win.

Aggression. You need to spread swiftly as Advent, and your cap ships' higher ratio of shields to hull means you can and should do just that. Your early access to culture techs mean you can spread out further than your rivals and get better income, thanks to loyalty boosts. A far-flung empire also means that while you get later access to trade, it's better when you get your long route going.

Preservation. As Siddy points out, your fleet mechanics are built around preserving your ships. Also, the Advent fleet benefits very well from strike craft, which are free to replace. Trading strike craft for enemy structures and vessels which cost money is a good thing.

Loyalty affects tax income.  Make sure to build broadcast stations and build extra in worlds far from your capital.

Yes, as the Advent culture and cheap firepower is your friend. On small maps I'd build just enough disciples to be able to take weaker neutral planets, or to help caps clear them faster, and then try to get 3 military labs up in enough time to get some carriers and illums out. Let the pirates come to you, which should be a breeze with them. You can then decide weather to go on an all out attack or just raid their economic buildings and withdraw if serious resistance is encountered.

On larger maps, you can start with some civil first. You should get two civics out in the first few minutes to get culture spreading, and then as you acquire more worlds, get to four for trade ports and resource focus. Now, as the Advent does not have refineries, you have to get the most out of trade ports. You can build as many of them as you have logistical slots for, and devoting an entire planets logistics to them can have an astounding effect on your economy. If you need credits, just build them and let them go. If you need metal and crystal, its best to build trade ports on volcanic and ice worlds respectively then use resource focus. It does seemingly little at first, but you can see a real boost when its fully upgraded, giving a 25% bonus to asteroid collection in the planet's gravity well. Sense this bonus is stackable, if you can spare 16 logistical slots, you can double your rate of collecting resources. As it's a percent bonus, you get the most impact when you have high allegence, so worlds close to your home with a culture bonus (preferably after getting the max allegience to +20%) will be far more productive. While this may not equal the bonuses that the TEC and Vasari get, the fact you can switch from credits to metal/crystal at will allows you to be better able to deal with a particular scarce resource, and should allow you to compete in the same league as the other economies.