As soon as the mission begins, run to the covers to attack the Weyland-Yutani mercenaries. Don't lean out when you don't have to, but make use of the exploding barrels.
When the fighting is over, check the area for collectibles (there should be a full set or armor somewhere there), then follow your teammates towards the building pictured above. Don't hurry, because you won't be able to go inside until you've shut down the enemy sentry turret that's facing it. Disable it, then open the door using the welding torch.
Take the newly unlocked passage. Depending on preferences, you can pick up the M240 Incinerator Unit (the above screenshot) or stick to your current weapons. Personally, I recommend taking the flamethrower; you'll only be facing Xenomorphs for a while, and the flamethrower is very effective against them.
Go through subsequent linear corridors; you won't run into anything or anyone for the moment. Stop when you get to place shown above, with the alien eggs. Starting from here on, make sure to take a moment and destroy the eggs from a safe distance; you'll save yourself the stress of having to worry about defending against Facehuggers (if that happens, rapidly tapping tap the interact button).
Be cautious, because apart from Facehuggers you'll also come across "adult" Xenomorphs. Given the presence of Lurkers, caution is a crucial factor. Try to prevent any face-to-fact encounters, all the while keeping an eye on the motion tracker and listening to the sounds of their presence.
Keep moving forward through the egg-infested area to reach the above ramp, which will take you to the upper gallery. The game will give you another objective.