Starting out
Use your initial workers to start constructing buildings and to start harvesting Lumber. Create additional food for your new army. Build a Hero and your first offensive units via your first "Barracks" building. Once you have built an exploration group with a Hero, send it out and kill nearby Creeps. Do not send your Hero out alone or with only a few units. You need to be able to take out at least 3 Creeps with ease. This will help you level your Hero up, find useful Hero Items, and generate a little additional Gold.

Town Portal Scrolls
It is very important to know all about Town Portal Scrolls.

Defense
Defense can be very important. Build Tower Buildings (Orc Burrows, Guard/Cannon Towers, Spirit Towers, Ancient Protectors, etc.) near your Gold Mine and at Expansion Towns. Sometimes if you lose a battle, your town can be lost too because you do not have enough units to defend your town and enough time to train more. Towers can give you added time to rebuild your army and sometimes drive off a significant enemy attack. Often Towers are enough to make players retreat from attacking your town or Expansion Towns. Be careful not to focus too much on defense, because it often comes at the cost of offensive units. Generally to make progress, you must be continually attacking the enemy. Surround your Towers with buildings to block melee characters from reaching them.

Food
To create a new unit, you must have enough food grown. Be sure to build enough Farms, Orc Burrows, Ziggurats, or Moon Wells so that you will be able to create units when you need them. At the top right portion of the screen, to right of the resource counters, is a listing of Food Used/Food Grown. Keep a watch on that indicator and get more food as necessary so that you have enough. There is no penalty for using more food than you have grown, but you will not be able to create any more units until more food is grown or less food is used which usually means some units die.

Food Cap
The total food cap is 100 regardless of the race you play. You begin the game with a ready supply of food courtesy of your Town Hall building. Each race has a special building that can be built to increase the food cap in regular increments. The Human Alliance builds Farms; the Orcish Horde builds Burrows, the Undead build Ziggurats, and the Night Elves build Moon Wells. However, no matter how many such buildings you create, you cannot exceed 100 food.

Freeing Up Food Space
Sometimes, you run into a situation where you need to quickly build specific combat units but you do not have food. In desperate situations, kill some workers or the most useless unit you can find to free up food for units that would better address the situation at hand. While this may seem an extreme solution to lack of food, it can help out in dire situations.

Corpses
When most units are killed they leave a corpse. Various spells and units can interact with corpses. Flying units and Heroes do not leave corpses. Corpses left on the ground last for 88 seconds before they disappear.

Know How to Counter Each of Your Opponent's Moves
No matter what your enemy does, you should know what to do. That's not easy to do but with research and experience it can be accomplished.

Recon Is Key: Know What Your Enemy Is Doing At All Times
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Watch Game Play Replays: Know Why You Lost
Warcraft III is a game of learning. When you lose you should know why. By knowing why you lost, you can find areas to improve upon on in future games. Watch game replays to figure out where you went wrong and look at areas in which you can improve. If you are stumped, give Replays to your friends and they can give you advice.

Offense vs. Defense
The best defense is a good offense. Don't spend too much time building defenses for your town. A huge defense may protect you from attacks but to advance in the game you need to attack the enemy town and prevent them from expanding and producing. Instead of building too many defenses, you will have much more success building more offensive units, expand to more resource spots, upgrade, and attack.

Clearing Trees
On certain maps you may need to clear trees away to reach Gold Mines. There are also situations where an enemy has heavily defended the front part of a Gold Mine while the backside only has trees for defense. In situations such as those you need to clear trees away.

The obvious way to clear trees are:

Humans: Mortar Teams
Orcs: Demolishers
Night Elves: Glaive Throwers or Ancients' Eat Tree
Undead: Meat Wagons

There are also other options for clearing trees:

Humans: Flame Strike
Orcs: Earthquake
Night Elves: Force of Nature. This is an ability that is available right at the start of the game. Use it as a Night Elf advantage to clear your way to safe expansions or to clear a path to an undefended part of an enemy town.

Mountain Giants! - First have a Mountain Giant grab a tree (which clears away a tree) and THEN attack trees with the tree you just grabbed. This will clear out the trees! Once your tree weapon has been used up, grab another one!

Undead: Death and Decay, Infernal - Target the Infernal on a group of trees you wish to clear. Ghoul - Select Attack and select the tree, do not right-click on the tree.
Neutral: Buy a Goblin Shredder!

Special Anti-delay Tactic Rules
If you (or your entire team in a team game) lack a Town Hall building or equivalent for 2 minutes, you are revealed to your opponents as if you had share visioned them. Partially completed Town Hall buildings count as Town Halls.

Unit Speeds
Units can never move at a speed slower than 150 (roughly 60% the speed of a Footman). This means that stacking slow effects such as Slow Poison, Shadow Strike, Slow, etc. will not virtually stop units.

Random Race Selection
Selecting Random Race allows you to have a slight advantage over other players. When you have selected random they don't know exactly how to plan their Heroes or strategies around your race. Some races are easier to attack early in the game than others too and the enemy won't know this information (unless they scout). Enemy players are forced to come scout you to see what race you are so they can modify their strategies. It can be a challenge to play Random because it requires you to be able to play all four of the races effectively. You shouldn't know only two or three races then lose games with the remaining races. You may also get the same race (the one you're weakest at) several times in a row. While some players think this is a bug this is actually what random means. Each game you have a 25% chance (confirmed through stats) of getting each race. But there is no history of what races you just played so your chance to get the same race again is again 25%. Once you've played Orc you do not receive a smaller chance to play Orc the next game. So sometimes you get the same race again and again. This is how random works! Because Random is the most challenging besides Tournament Random, typically better Icons are given to random players. Of course Blizzard's idea of better might differ from yours.