In classic Tomb Raider combat two basic rules apply: Pick the right weapon for the job and conserve ammunition.
Early in the game, you probably won't have much choice—pistols, maybe a shotgun. Later, you'll have a variety of weapons at your disposal so you'll need to set some priorities. You may be tempted to use your high-powered weapons as soon as you find them, but remember that the enemies in the later levels are bigger and tougher. Also, certain enemies can be taken out more easily with certain weapons. I've provided some general tips below and a few case-by-case recommendations in the walkthroughs for each level. You'll undoubtedly come up with your own methods.
SHOOTING FISH (OR WHATEVER) IN A BARREL. Most Tomb Raider enemies can't climb. So creatures on the ground can't reach you if you climb on a block or ledge. Similarly, most creatures in the water can't move on land, and vice versa. So get Lara to a spot where her enemies can't go, then use the pistols, which have unlimited ammunition, to make the kill.
Naturally there are few exceptions. Obviously if someone or something is shooting at Lara, pick a more powerful weapon to fight back. Also, human foes can shoot at Lara when she's in the water, so you'll have to come out to finish them off.
USING LARA AS BAIT. If you suspect there are enemies below but they haven't come out yet, you can make Lara drop and hang to draw them out. This is the same as the safety-drop move, only you don't let go. Instead, hang until the enemies emerge, then pull up and take care of business. This works with water dwellers like crocodiles and barracudas too. Take a quick swim to get their attention. Then climb out of the water to shoot them.
HANDLING HUMAN ENEMIES. In TR1, there are times when the human adversaries won't die no matter how much you shoot at them. Pierre Dupont can't be killed until the end of the Tomb of Tihocan, and Larson won't die until the end of the Sanctuary of Scion. So when you meet these guys earlier in the game, shoot with pistols until they run off or a cut scene starts. In Natla's Mines, however, you'll need to dispatch some human foes quickly, so go ahead and use the best weapons you have.
In TR2, all the human foes can be beaten. I've found pistols or automatic pistols to be the weapon of choice except when fighting heavily armed or multiple attackers. In the water levels of TR2 and 3, the scuba divers can be taken out in the water with the harpoon gun, but it can be difficult to aim, and the slow reload time doesn't help. For divers and other underwater enemies, it's often easier to get Lara to dry land as quickly as possible then use the pistols.
Also in TR2, you'll have a chance to make a few friends or at least avoid making enemies. In the Barkhang Monastery level, do not shoot at the monks. If you do, they'll dog you through the entire level, making things much more difficult than necessary. If you treat them with respect, they'll even help you fight the cult goons. Likewise the ice men in the Golden Mask expansion: They'll help Lara if she doesn't harass them. There are also places in TR3 where holding your fire will help you advance in the game. The walkthrough describes them.
WEAPONS AND TARGETING IN TR4 & 5. Lara's arsenal increases in every game, but in The Last Revelation and Chronicles, there are even more options. Some weapons can be combined with the laser sight for more accurate long-distance targeting, and some can use different types of ammunition. Details on combining items and choosing ammo appear on the TR4 and TR5 controls pages, along with information on automatic and manual targeting.
UNDEAD CREATURES IN TR4. Early in this game there are some enemies that won't die no matter what. In most cases, it's because they're already dead (e.g., mummies). So don't waste precious time and ammo blasting away at something that won't go down permanently. If you can't figure out how to get around these nasties, check the walkthrough. (If you are playing through a second time or have used the all-weapons cheat, you can destroy mummies and the like with explosive weapons.)
Some screenshots courtesy of Katie's Tomb Raider Site.