When fighting the Combine, or the zombies, there are many ways to defeat them. Here are a few efficient ways to get rid of them.

Combine Metro Cops:
Most efficient weapon(s): Any
Distance of engagement: Close to Medium range
Notes:
They are the basic grunts of the Combine forces, and are easily killed, they use pistols and the standard assault rifle (not the Combine one). They also use Manhacks. They are moslty encountered during chapters 1 through 5.

Combine Soldiers:
Most efficient weapon(s): Combine Assault Rifle, .357 Magnum, Crossbow, Gravity Gun
Distance of engagement: Medium to Long range
Notes:
These soldiers use various weapons such as the pistol, assault rifle, Combine assault rifle and grenades. They also have Manhacks. Appear in chapters 7 onwards.

Combine Overwatch Soldiers:
Most efficient weapon(s): Combine Assault Rifle, .357 Magnum, Crossbow, Gravity Gun
Distance of engagement: Medium to Long range
Notes:
The stronger soldiers first met when approaching the Nova Prospekt prison facilities. They use the same weapons as the Combine soldiers. Appear in chapters 9 onwards.

Combine Elite Soldiers:
Most efficient weapon(s): Combine Assault Rifle, .357 Magnum, Crossbow, Gravity Gun
Distance of engagement: Medium to Long range
Notes:
They are the toughest Combine soldiers to deal with. All are equiped with the Combine Assault Rifle, and they are very efficient fighters. They use the Assault Rifle's secondary mode too (the energy orb). They are mostly encountered from Chapter 11 onwards.

Now for the headcrabs and zombies:

Normal Headcrab:
Most efficient weapon(s): Any, even the gravity gun's first firing mode can kill them after a few blows.
Distance of engagement: Close (they are always jumping in your face!)
Particularity: Creates Zombies

Fast Headcrabs:
Most efficient weapon(s): See Normal Headcrab
Distance of engagement: See Normal Headcrab
Particularity: Creates Fast Zombies

Poison Headcrabs:
Most efficient weapon(s): Long distance, high damage weapons.
Distance of engagement: Medium to Long range
Particularity: Has a very distinctive screech before it attacks. Injects a neurotoxic agent into your bloodstream which leaves you with 1 point of health. Creates Poison Headcrab Carrier.

Zombies:
Most efficient weapon(s): .357 Magnum, Assault Rifle, Combine Assault Rifle, Gravity Gun, Shotgun
Distance of engagement: Beyond claw reach (medium)
Notes:
Always aim for the head of the zombie. If you kill the host body, the headcrab will come off and attack you.

Fast Zombies:
Most efficient weapon(s): .357 Magnum, Shotgun, Combine Assault Rifle, Gravity Gun
Distance of engagement: They never stop until they are in your face, so engage them as soon as they come in sight.
Notes:
Same as zombies, aim for the head, use a the shotgun's double shot firing mode when the zombie is getting too close. Be careful when many are coming towards you.

Poison Zombies:
Most efficient weapon(s): Combine Assault Rifle, Rocket Launcher, anything which explodes (barrels) or which set things on fire (butane gas bottles).
Distance of engagement: Long range (the zombie can throw the headcrabs at you quite far).
Notes:
Take care when engaging then, as they will throw the poison headcrabs in your direction. They should be engaged as soon as possible, to prevent them from spreading their headcrabs. Anything explosive is tremendously efficient, as it will most of the time kill the host, leaving the headcrabs in a small pack on the floor. If you are fast enough, you may launch another explosive at them and kill them all in one go. Otherwise use the shotgun, but this is risky.

Other Combine encounters:

Combine Gunship:
To defeat them, use the rocket launcher and guide the rocket in erratic patterns. This will ensure that the rocket reaches the gunship before it shoots it down. It may take a bit of practice to steer the rockets, but you get the hang of it pretty quickly.

Combine Striders:
Unlike Gunships, they don't shoot down rockets, so simply shoot the rockets straight at them, steering it when there is an obstacle.

Combine Dropships:
Can be shot down, but usually this is a waste of ammo.

Combine APCs:
Usually they don't bother you much, so they are pointless to attack.