Okay, I just opened up the map in GalaxyForge to get a look.  First things first:  any time you see a string of asteroids near your homeworld, they are your top priority.  Asteroids are inexpensive colonize but nearly as lucrative as a full-sized planet.  Those asteroid belts will also contain a random amount of neutral extractors; those have higher efficiency than normal extractors in a gravity well and are also high priority.  It should be noted that you don't need to send a capital ship to take any of these; colony frigates with a few combat frigates for backup will suffice... at least so long as your enemy doesn't attack.

Dead asteroids are nasty business.  They offer zero economic benefit, but still cost you in the form of infrastructure upgrades.  Fortunately the AI doesn't know how to abuse this, but a dead asteroid next to your homeworld is actually a huge liability, because a smart player will set up a staging base there.

On such a small map, it's unlikely the pirate base will enter into the actual equation.  Although risky, you can attempt to run through it and hope you don't take too much damage.  Wouldn't recommend it unless you're in desperate need of a short-cut.  The AI will not attempt this.

The challenge on this map is that your three different approaches to attack the enemy are not connected.  If your forces are at his dead asteroid, there is no way they can back-track to defend your own outer asteroids if they come under attack.  As a result, you will need multiple strike forces.  One big fleet won't cut it.  Now, the AI will always keep one big fleet.  To beat this, you need to have defenses (particularly repair bays) set up.  Whenever he comes to defend himself, retreat back to your repair bays (your other force will be attacking at this time).  If he follows to attack you, you'll need to use your defenses to survive.  If he turns around to attack your other fleet, just return right back to the battlefield and start attacking him again.

My approach on this map would be to immediately send my colony capital ship to my enemy's dead asteroid and start setting up defenses there.  In the meantime, I will scout the desert planet, and if it has few militia I will capture it and use it to build a forward frigate factory.  While capital ship is occupied on this offensive move, I will send a few frigates to capture my asteroids as well as my neutral extractors.  Once I am in position, I will attack the enemy homeworld in an effort to grab his attention.  I will then build a second capital ship to attack one or both of his flanks.  The AI is very poor at handling these two-pronged attacks and will likely collapse shortly afterwards.

Most of the time in the early game, you rely on selling metal and crystal to the black market to generate credit revenue.  This is normal; an extractor economy is easy to get running and the cost of getting something more elaborate is often too high.

Actually, asteroids (dead and otherwise) have exactly the same amount of tactical logistics space as a full-sized planet despite being much smaller.  This means you can actually build much denser fortifications here.  The best defense is still a bank of repair bays with a strong fleet holding the location, but that's all the less area that fleet has to protect.  In a big gravity well it's entirely possible for an enemy to completely go around your fleet (mind you, an AI won't do this).

If you're going for imperial labs, you may as well try to slowly colonize the inner worlds.  You can focus on the planets with low militia count first, then when you have the opportunity send a capital ship to clean up the harder ones.

Okay, a few things

1. You probably waited too long to take your asteroids if you're doing it with LRM.  Three Cobalts (or three Skirmishers or four Disciples) will cut it.
2. More important than the number of extractors on the planets you scout are the number of militia.  This will indicate how easy or hard the encounter is.
3. Make sure you get a frigate factory up and running on that desert.  You know how far it is from your homeworld, even cutting one jump out of your journey will save you much needed time (you can even set the rally point from your factory to another gravity well.