This fight is a bit of a tough one, because of the more confined spaces the Covenant forces will be much quicker to engage you and in greater numbers, and they have the support of two Banshees to start with.
As soon as the fight starts, make a beeline for the first small building to your left. Inside, the Covenant are already there. Pop the eight or so Grunts with head-shots and toss some grenades if you like. Quickly take cover before the Banshees pepper you with plasma, they can easily take you out in open ground.
The Banshees are going to be the reason you're gonna have to hole up here for a while. You can try to go out, but unless it's a short sprint from point A to B with zero foes, they will always get you. So stay inside and get comfortable, fortunately you've got plenty of supplies.
The Covenant will attempt to retake the building, but for some reason they almost never use the ramp in the back. They will either funnel through the 2 doorways or go on top of the structure and just stand there doing nothing. The waves usually consist of either 2 or 3 Grunts, or a single Elite Major or 2 Minors. Anybody not going through those doorways are going to stay on the roof. (Note: Disabling the power core for this building is going to trigger 4 Jackal snipers to automatically appear with the regular waves, so make sure you time it right. The periods between the waves are long enough so that you can rest up and refill weapons with plenty of time to spare.
When you feel like all the waves are down, it's time to defeat the Covenant on the roof. Move up the ramp and peek out to get a good shot at the 2 or so Elites and a handful of Grunts that stayed up here. The Banshees will immediately take a shot at you, so be quick to dive back into cover.
When Palmer alerts you that Parg Vol may be in the area, this is a cue that most of the foes in this section of Galileo are gone. NOW you can deal with those annoying Banshees. Grab a Railgun and wait for them attack.
For all their potential power, the Banshee AI is very predictable. It flies around a small airspace with loops and turns, then starts a slow, straight attack run and fires about 6 bolts at you before turning back around to repeat the process. As it's making it's run, this is the best chance to charge up a Railgun shot and fire. The longer you can charge, the more power the shot has. The Banshee will make no attempt to alter it's course and will attack as normal no matter how many times you do this, although it will take many shots (about 6 "quick shots or three fully charged shots) to bring them down. As a bonus though, the Banshess get themselves stuck very VERY easily, so if that happens they're practically begging to be shot out of the sky.
Finally, the first section is clear. Move on to the bigger building through the small doorway and you'll meet two Elites with a Jackal or two. Use a Rail shot to kill them or toss in a grenade, it doesn't matter. That's all that you'll have deal with for now. The third and final power core is guarded by about four or five Elites, but the small building across from them has plenty of weapons to kill them easily. Since they don't even stay alert for very long, you can easily move out of cover and kill one or two then pop back inside and wait a few seconds to catch them unaware again. Once all of them are gone, tag the last power core and retreat back into the big main building. This and the smaller room with the power weapons are going to be your main points of defense for the rest of the level.
You're instructed to activate some defense turrets to aid Galileo, but then after they are turned on you have to deal with the reinforcements. It's several Grunts (about twelve or so), with two squads each landing on the rooftops of the turrets, consisting of 2 Elites and some minions each. Retreat from the roof tops back into cover so they don't swarm you with Phantom fire, and deal with the twelve man Grunt squad first, but be careful not to poke outside too much- the squad closer to you will take notice and plenty of the Grunts are carrying Fuel Rods. When all the foes NOT on the roof are gone, carefully move up the ramp to the nearest rooftop. Chuck some grenades and use your best weapons for point blank combat with two Elite Majors and 3 or so Grunts. The next rooftop will be in plain sight from here, although one of the Elites may have a Fuel Rod and will take shots at you. Go back below and grab a sniper rifle, and once most of those foes are cleared out, the last wave will appear. It's a doozy, containing 2 Hunters, about four to six Elites, and a bunch of Fuel Rod Grunts. Swap back for your Rail gun if you still have it, if not take a shotgun or rocket and wait for the Elites to come inside the big building from multiple angles. Blast them, but beware as the Hunters will be coming in too and may show up at any moment. You don't want to get boxed in traveling the narrow hallways of the base. Deal with the Hunters as they come, hopefully in a more open area so you can get to their backs, and if it gets too hot remember the building housing the power core has more rockets if you need them. Once their down, take out any little guys left, and you're done.