Using Maneuvers
Maneuvers are going to be the key to maximizing the effectiveness of your soldiers in combat. They're perfectly capable of killing things on their own, but when you place them into a maneuver position, you'll find that their lethality will be greatly amplified. With them, you can order your squadmates to use weapons that they normally won't. Although Sev and Scorch will occasionally use the sniper rifle or grenades, respectively, this is very rare; your teammates will almost always use their regular blaster rifle, no matter what the weakness of the particular enemy you're facing off against. Thus, you'll want to use maneuvers to help adapt to each situation.
One important note about maneuvers is that enemy grenades can cause your teammates to run for cover. When this occurs while they're in a position, they'll often not automatically return to the position, forcing you to disengage them and order them back in before they return to their task.
Sniper Position: Sniper positions are generally going to be the most common and the most desirable maneuver position in Republic Commando. When placed in a sniper spot, your teammates will become true engines of destruction, rapidly sending out devastating sniper rounds towards every enemy within view. Although some enemies, like the SBD and droideka, will be killed more quickly when your squaddie is in an anti-armor position, but sniping is never a bad choice, and indeed will rip through grunt-level enemies in one shot.
Grenadier Position: When a soldier is in a grenadier position, he will throw out context-sensitive grenades every seven or eight seconds: when there are droids in sight, EC grenades will be chucked, while organic enemies will get thermal detonators or sonic grenades. (Note that they'll call out "Thermal det!" no matter what kind of grenade they're throwing.) This can be a big help against large groups of enemies, but you'll need to stay clear of the area of effect of the grenades, lest you want to wind up caught in the blast. Note that the large targeting circle that appears on the ground when you look at a grenadier spot isn't the only place that grenades will be thrown; the grenadier can track enemies and attempt to throw grenades at enemies that have moved out of his designated targeting range. Indeed, you'll need to be careful, as your grenadier will sometimes overenthusiastically attempt to hit enemies behind cover by throwing grenades at the wall or other obstacle behind which they're hiding, which can sometimes reflect them back onto your party.
Anti-Armor Position: Anti-armor fire can devastate droid enemies, but these positions are unfortunately one of the more problematic in the game. Although your teammates can deal a lot of damage with their anti-armor attachments, you'll need to babysit them in order for them to reach their full effectiveness in these positions.
The main reason for this is that, whether by design or due to a bug, your teammates will not fire as soon as they've reloaded their weapon while in an AA spot. In fact, they'll wait five or six seconds before refiring, greatly reducing the amount of aggregate damage that they deal, and leaving them open to fire from enemies. If you want to increase their rate of fire, then place them in position, watch them fire, then wait for their reloading animation to end (it consists of them going to one knee, shoving another round in the weapon, then standing back up). After all this is done, order them to leave the position, then immediately tell them to get back into it. When you do so, they'll fire again immediately, allowing you to repeat the process over and over again. They'll easily fire twice as often as they do normally when you do this, allowing you to clear out enemies without having to use your own precious anti-armor ammo.
Turrets: Now we're talking! Getting your soldiers into turrets will let them blast away at incoming enemies with an unlimited amount of rockets, laser ammo, or what have you. In most cases, you'll have to remove the current occupant of the turret before you're able to place one of your own soldiers inside it; this will usually require a quick sniper shot to the head, as blaster fire is often going to be deflected by the energy shield protecting most turrets. Of course, said energy shields don't protect your teammates very well, and in most turrets, they'll be just as vulnerable to fire as if they were standing around in the middle of an open area, but this is made up for thanks to the immense firepower of turrets, which can usually kill off enemies before they're able to return fire.
Door Breach / Door Slice: There are a lot, and we mean a lot of locked doors in Republic Commando. When you face off against one, you'll often be given a choice between slicing it (hacking the lock open) or breaching it (placing an explosive on the door and blasting it open). There are pros and cons to each method.
When you breach a door, your teammates will take up positions on either side of it first. When the charge goes off, they'll throw a grenade into the room beyond and move in, ready to fire. The primary advantage of this is that it's quick (it can be done immediately, while slicing takes ten seconds) and that you get an essentially free grenade in the bargain. The drawbacks are that all of the enemies on the opposite side of the door will be alerted to your presence, which will shift their positions in the room and bring them closer to the door than if you had sliced it. You'll also be unable to command your soldiers for a second or two after the breach while they're moving in, so if you see a sniper position on the far side, you'll have to wait a second or two before you can set a soldier there.
Slicing the door is generally going to be the better alternative on hard mode, as it won't alert any of the enemies on the far side of the door, and you'll be able to proceed in on your own pace, instead of having to follow your soldiers in. Although you miss out on the free grenade from a breach, and you will have to wait ten seconds for the slice to go through, the extra control you gain over your passage through the door is usually going to be worth it.
If you want to see which flavor suits you best, try loading up a level of the game and going through it once with all breaches and once with all slices and see what the differences are. There are some doors that cause turrets to activate when you breach, but remain inert when you slice, and so forth; a lot of these differences depend on the specific door and situation.