Generally speaking, you have two choices for opening capital ship: the Progenitor and Halcyon. The Progenitor a long-term option, giving you a stronger economic foundation and better supporting large fleets. The Halcyon is a military opener and your strongest combat capital ship by a significant margin. It also remains relevant into the late-game and is very effective against militia, so in general most people err on the side of the Halcyon if they're unsure of which of the two to go with. On bigger maps, use the Progenitor, smaller maps focus on the Halcyon.
General advice for any faction is to focus on early colonization. Building a strong foundation for an empire is going to pay off marvellously even in the short-run. The Progenitor can be used to "skip" past planets whose militia are too time-consuming to deal with, allowing you to beeline for key tactical worlds. Against the AI, it's actually quite easy to box him in with 3 or 4 planets within the first 20 minutes of the game, then backfill all the worlds you missed.
For colonization, try knocking out the Krosov Siege Frigates in the militia first, then colonizing the planet. The militia can no longer bombard the planet, and you can build a defensive turret to help clean them up. From there, you can keep fighting to gain the XP, or keep moving and colonize planets. A lone colony frigate can actually colonize an asteroid in this manner. Just move and colonize it and start construction of a turret right next to the planet. It will complete and destroy the militia siege frigate before it can finish bombarding the world. Using such minimalist tactics, a small group of frigates can colonize worlds with low militia levels while your capital ship pursue more heavily defended or remote locations.
Avoid unnecessary expenditure. Keep light on technology, minimalize planet upgrades, and focus on purchasing things that directly increase your income. Extractors, trade ports, maybe a bit of culture. Stay away from stuff that says +7% income; this early small numbers like 7% are a drop in the bucket and not worth your time unless you really don't have anything better. Usually you want to move up to Illuminators or Destras as your primary combat unit. Disciples are passable, but too easily beaten by the Javelis LRM and Kanrak Assailant. For defense, get the repair platform operational as early as possible. This thing is a real life-saver, especially for those fragile Advent capital ships.
For economy, your best option in the long-run is a big chain of trade ports. The longer your trade route, the more money each individual trade port makes. Breaching 100 credits per second income within 30 minutes is totally doable. Avoid "resource focus" like the plague. This is ability is downright horrible and you will pretty much always get more money from trade port mode.
A good fleet combo is the Iconus Guardian with the Progenitor Mothership. Use the Progenitor to keep shields topped up, while the Guardian absorbs a lot of the damage. If you add in a Rapture Battlecruiser with Vertigo and max out your shield mitigation bonuses, this can be a crazy defensive combo that brings you to ~99% overall damage reduction, which is all but invulnerable to anything that doesn't bypass shields.