Playing the zerg, you have to remember about the unusual way of building, increasing population, and training units. Drone, the worker, is responsible for transporting materials and erecting buildings. It doesn't create them in a standard fashion but mutates and thus becomes a building.
Although you lose a worker in the process, some zerg buildings are cheaper than those of their enemies. For example, Hatchery costs only 300 minerals instead of 400 in the case of a Command Center/Nexus, while zerg refinery costs only 50 minerals instead of 100 like the ones of the terrans and protoss do. Because of this unusual strategy, you need to have more workers than the remaining races. Moreover, every building aside from Hatchery, the main zerg structure, have to be placed on the special goo called the Creep. It is spread by your Hatchery and its improved versions as well as by watch Turrets - Creep Colonies. These can be later upgraded to guard Turrets. Buildings can regenerate so their hit points are restored automatically.
Training and producing units is also completely different than that of the terrrans and protoss. Hatchery creates up to 3 Larvae that can evolve into different units. To create various kinds of troops, you need technological buildings that unlock the genetic code and allow the Larvae to mutate into, for example, Zerglings or Hydralisks. No other building supplies Larvae and you can't train units any other way, but you can acquire some by the second evolution of the already existing troops. This is why it's a good idea to build more than one Hatchery to speed up production. In the later missions, it's worth it to have up to 2-5 such buildings. If you click Hatchery, you can use a button that selects all available Larvae. If you have more Larvae, you can double click a Larva to highlight all units on the screen and then create an entire group of units. The zerg can create a grand army within a very short timeframe.
Overlords are mobile air units. They have several properties. They have the Control factor, which is the equivalent of Supplies and Psi. This factor decides how many units you can control and increases the population limit. They are Detectors, as well, which means that they detect invisible units. They only 100 minerals and have a lot of hit points. With time, you can improve their speed and visual range. The last enhancement is the possibility of transporting your armies. It is the only zerg unit that is capable of carrying out landings between islands or other inaccessible areas. Overlords aren't included in the population limit so it's a good idea to have more of them than you actually need, given their numerous properties. Nydus Canals is a situational zerg building. When you build the first one, you can select it and create another one in any place on the map as long as the Creep is there. Having created such a pair, a link between them is established and you can use it as a kind of a teleport. Units move between the buildings instantly, which can be used both offensively and defensively.
The zerg rely mainly on fast and cheap units. You will use the same kind of troops in the majority of the campaign missions. With time, you'll gain access to more specialized units and unique support troops. All zerg units and buildings can regenerate and are restored to full health, given time. This process is faster if they are situated on the Creep that comes from the main building and Creep Colonies. Most land units can Burrow which makes them invisible and speeds up their regeneration. It's a good idea to take an Overlord for every operation. It can detect invisible hostile units and can transport your troops.
For most operations, you'll use the basic combat units - Zerglings and Hydralisks. The former cost only 50 minerals, and a single Larva gives you two units (each using half a population point). They move very fast and later their attacks become faster, too. Their weaknesses include few hit points and poor melee attack. Hydralisks are a great addition for your land forces. They are more durable than Zerglings and, what's more important, can attack at a distance. You can increase their attack range in their technological building. At later stages of the game, you will be able to train Ultralisks, which are powerful units that look like battle elephants. They have 400 hit points and attack enemies hand-to-hand on a small area in front of them.
Zerg air force is poor compared to that of other races. Mutalisks are fighters that attack land and air targets. They have low attack value but their projectiles bounce off the enemy units and attack several buildings/hostiles in the area. Try to have a numerous group of them because they stand no chance against Wraith or Scout in 1 vs 1 combat. Later, Mutalisks can evolve into the very useful Guardian bombers. These have excellent range and powerful missiles. However, they can only attack ground forces. This is why you still need Mutalisks as protection against hostile aircraft.
The last group of units in the zerg army are support units. They can't attack directly but have incredibly useful abilities. However, they use Energy, an equivalent of Mana in other games. A Queen is the first "wizard" you'll receive. They can infect a hostile unit with the Parasite ability and thus have a view of its surrounding until death. Ensnare slows enemy groups down. The most useful ability is Spawn Broodlings. The Queen can use it on an enemy land unit, which destroys it immediately, and spawn two Broodlings (weaker version of Zerglings) in its place. It is very useful in eliminating powerful land units such as Siege Tank or Ultralisk. The only units you can't target are Probe, Reaver, and Archon, all of which belong to the protoss. A Defiler is a zerg creature that resembles a scorpion. It can Consume a hostile unit to replenish its energy bar by 50 points. Plague can be cast on a group of units/buildings of the enemy and takes up to 300 points of health. However, enemy units/buildings cannot be destroyed in this way (but then, what's the problem with destroying an enemy with 1 hit point). Dark Swarm is a defensive ability, although it can be used offensively with time. A special swarm covers a small area. Units within this area are immune to hostile ranged attacks. You can use this ability to protect your army or while, for example, assaulting terran Bunkers with Zerglings.