Most of these missions are stealth-based, so this is easier than the other Veteran achievements. During Stealth parts, stick to your partner and listen his orders. I'll cover a few troublesome areas with tips and recommend which guns to use. If you want a video, there's nothing better than the ones in the official walkthrough.
In "Eye of the storm," hold on to your sniper, even after the stealth part is over. There's a part where you'll need to take out two turrets. Lose the Pistol for an AK-47 or Similar ASAP. Later, there's a part right before the end where you get ambushed. You'll know it because a civilian will run into the enemy in front of you. Take cover on the right side of the room before. Soap should run in the room. It's easiest to clear the room using flashbangs.
In "Blood Brothers," Hold on the weapon Price hands you, and remember its grenades. Swap your pistol for an assualt rifle with a good scope early, and look out for a machine gun later on. One can really help you in the park area. Throughout this level, it says to protect Soap and Price, but they can take a lot of punishment. Focus on keeping yourself safe.
After you give Soap to Price, you'll pass through a store into a courtyard. Stay in the store until Price opens the next door, clearing the ground troops, then sprint across the left side.
In Stronghold, one of the first two guys you take out has a suppressed AK-47. Trade your pistol for it and be looking for a P90 to upgrade to. During the part when the enemies are using flares (after the nightvision silent kills) use the P90 and focus on advancing.
In the courtyard, after you past the two suits of armor, go right and up along the ridge. It might take a few tries, but it's much quicker than going down. Use your flashbangs before rounding corners and the P90 to clear the clumps of three enemies.
In the main area with the vehicles, don't bother with the enemies in cover or the tank, just run along the back and right to the jeep. Hide behind the cover halfway there and let your health regenerate. Once you're in the jeep, you're unkillable; just remember to open your chute.