Destroying the Hyperspace Inhibitors
You'll come out of Hyperspace a short way away from a large Vaygr Command Station; there are some Hyperspace Inhibitors attached to the three large asteroids between you and it. Situated by each of them are several Vaygr Missile Corvettes, Laser Corvettes and Lance Fighters, as well as a Carrier. You should bring some Pulsar Gunships along for this, so start building some and research the Pulsar Gunship hull upgrade, the Flak Frigate hull upgrade, and hull upgrades for all your Fighters if you've not researched it already. Speed upgrades should only be researched for Interceptors or Frigates at this point.
Move your Resource Collectors to start mining the nearby asteroids, then send your fleet to the asteroid on the left, looking the way your Mothership is facing. This will prompt all of the Vaygr forces out here to move towards you, so when you get there eliminate the Corvettes and Lance Fighters as they appear, then move on to the Carrier when it arrives. You're then free to destroy all the Inhibitors on that asteroid, but make sure to listen to Fleet Intelligence when it suggests you use Marine Frigates to capture the Hyperspace Gates beyond them; they will come in use later.
For this phase of the mission, you will want to tool up your fleet a bit; your Carrier needs Fighter and Corvette-class Facilities if it doesn't have them already, and your Mothership should receive a Capital-class Facility. This will enable it to build more Carriers, but don't for the mean time (they cost too much, and you can't afford to kit it out properly yet). If you can afford more things, invest in an Advanced Research Module and Advanced Sensor Array for your Mothership, although after this you will probably be low on RUs - if you have less than 3000 RUs by this point, don't bother, as you will need them in case you have to replace ships that are destroyed. You can do this in relative safety as if you've already wiped out two asteroids and the defenders and Hyperspace Gate there, the only threat will be the occasional group of Fighters or Corvettes that warp in using the remaining Gate - don't finish off the remaining Inhibitors as this will cause the game to advance before you're ready, something you definitely don't want to happen. You should have a fair amount of all ships by this point, including a few new Torpedo Frigates for more efficient Corvette and Frigate destruction. If you find you have an abudance of one type and not enough of the other, feel free to Retire some using the special commands box in the lower-left to turn unwanted ships into some RUs.
Taking Out the Command Post
Once the last Inhibitor goes down you will automatically initiate a short jump closer to the Command Station. Your aim here is to destroy the structure before it is fully-repaired - this is quite simple; send in your entire fleet and it will go down quickly. Once you've done this a Vaygr Carrier and Shipyard will appear off to one side of the former location of the Station, and will start constructing Corvettes and Frigates to deal with you. Take out the Carrier first by swamping it with your fleet, then deal with any smaller vessels that appear. Move on to the Shipyard afterwards. With a few Torpedo Frigates and Bombers this won't take long, and once you've destroyed it all you'll advance to the Gehenna asteroid field.