Only a couple of hings to add:

• You might wanna make your first cap the coloniser ship. If you don't, you'll be wasting fleet and resources on 2 colony frigates.
  You'll need 2 to replace a colonising cap because they have abysmal AM regeneration.
  If you choose not to make your first cap the coloniser, you'll probably want the full-on battleship. You'll lose early-game expansion speed advantage for mid-game firepower (your combat cap will be at a higher level).
  Which road you take is up to you - might wanna practice both as each is more suited to different games.
• As gingerbill mentioned, upgrade your planets max population cap fast. Until you do the 1st 1-2 upgrades, you'll be paying a heavy underdevelopment penalty.
  As a rule of thumb, do not colonise a planet unless you can afford to immediately upgrade it to profitable level.
• Its important to have a strong economy, as you've noticed. Pursue secondary income generators - trade ports, refineries, culture - to reap extra benefit for great justice.
  

Firstly, welcome;  I hope you enjoy the game.

Secondly, the AI cheats, even at easy so don't be surprised if he does a few things you are unable to equal.

Thirdly, income is always kinda "slow."  If you just need Credits in a hurry you can always sell Metal/Crystals for a boost (Pirates/Black Market Tab), for longer term you probably know from other 4x games what you have to do.  As the others have noted though, when you take a new planet make sure to upgrade the Infrastructure as soon as you can, without those upgrades fresh planets usually generate negative income.  You can hover over your current Credit amount for a more detailed listing on how much each planet is generating.

Build mines and a capital ship factory then build a colinizing capital ship and capture planets as soon as you have the mony to upgrade them to maximum population and build mines there.

Samurye.

Okay well there is a rule that I play by which gets my economy pretty strong pretty fast. Step one build cap ship factory and start on all the extractors, step 2 buy one hundred crystal from black market, research the planet infrastructure, step 3 build as much scouts as you have the number of phase lanes ex 3 phase lanes 3 scouts, step four while the cap ship factory is building click on it and que up the colonizing cap ship (trust me every second counts), while the capital ship is building waste money on building light frigates until your cap ship is ready, when the cap ship is built get the colonize ability and build a squadron of fighters. Send them to the nearest asteroid (trust me there is always an asteroid) defeat the siege frigate first and then colonize the planet or else be ready for a rude awakening. With the money that you saved up research the infrastructure and build all the extractors. Next if there is another asteroid nearby take it. If not keep your fleet at bay. This is where the valuable intel your scouts have gathered comes in handy. Zoom out not too much and look at the types of planets around your own, if there are Volcanic or Ice or both then build two civic stations and research the required tech, (while you're doing this try to build more ships as colonizing will require more effort) then attack with your whole fleet and take the planet/planet's, if there are no Volcanic or Ice planets and are Terran and Desert then build two military stations and research the LRF design (for Vasari you need 1 lab TEC 2 Advent 3), the military labs are necessary because these planets are better deffended and therefore harder to colonize, again build a considerable force and attack and take over the planet, and research all the infrastructure to avoid loosing money and build all the extractors (don't worry if you don't have enough money for upgrades just wait a little more and then build). Don't worry about defenses yet, on small maps I would capture 5-8 planets and then settle down and fortify on large maps I would get 15-20 planets and then settle down, practice on the small maps and get a feel for it and make sure that on the settings you have high resources locked teams and normal research and just to start off make your enemy an easy fortifier. Also if you're having trouble with pirates turn them off. But if you want pirates think of them this way, pirates are your friends you can bribe them to attack the enemy for a small price, but don't put bounty on the enemy anytime you want just on the time when the pirates are about to attack because someone else can kill the enemys ships and get the money. Hope this Helps and enjoy the game.   

I must say alot of people are saying build up a fleet. A single capital ship, which I would go for the colony capital ship at first, should be more then enough to take any planet near you, unless you start on the enemy home world. Don't buy any crystal or metal, but sell 100 metal, you usually don't use it the first couple minutes. Build up all your mines and taxes.