The more damage a gun (or class of guns) does, the more likely it is to be able to shoot through objects. That means that LMGs, assault rifles, and sniper rifles are better equipped to shoot through objects than shotguns, SMGs, machine pistols, or handguns. This doesn't apply to launchers, which inflict massive damage but are effectively useless for penetrating objects.
If you want to better shoot through objects, pick a LMG, assault rifle, or sniper rifle and use it a lot in order to increase your weapon proficiency level with that particular weapon. The more you use it, the more proficiency choices you unlock -- including one called
Impact, which gives your bullets increased penetration. It's effects are similar to Hardened in Black Ops, Deep Impact in MW, and Full Metal Jacket in MW2.
As for aerial weapons, things that deal bullet damage (the helicopters) are more likely to penetrate objects than the ones that deal explosive damage (a predator missile).
Ultimately, even with with the proper gun and Impact equipped, it is going to take some trial and error to get a feel for when you can shoot through stuff, as some objects are simply easier to penetrate than others, regardless of appearance. One example that springs to mind are the concrete pillars in the center of the map
Interchange; they are notoriously easy to shoot through even though they shouldn't be.