This page details the specific tactics of AWBW units. Advanced strategy may be counter-intuitive, so understanding the nuances of units beyond their stats is crucial for competitive play. (This page, authored by Nahbien, may be outdated, and is due for revision).
Infantry are the base of any army. You need Infantry to capture. Also look into using your Infantry to Meatshield your other more valuable units. Infantry are also useful in deterring a Mech Rush as they are the most cost effective way of stopping it. Infantry are also useful in Blocking a valuable property, chokepoint, or landing point. The enemy cannot capture your property easily if a unit is on top of it, even one as weak as an infantry. They will most likely have to use two other units to kill yours first, wasting time and valuable units. Having Infantry block a pass can cause units to not be able to pass completely, giving you Zone Control, or make them go around you and use precious movement points (and perhaps not be able to attack certain units, or attack from valueable defensive terrain). Infantry are useful at stopping opposing infantry from doing their own capturing, and at 1000 funding to produce, even sending one on a suicidal mission to delay an infantry from capturing is worth the cost. 
Mechs are a decent unit. Against an inexperienced player, a Mech Rush can prove very devastating. Aside from that fact, you don't see Mechs too often as they are generally too slow to be really effective. Mechs are basically a tank without the armor and movement, but the upside is that they cost very little. Mechs are good in packs where vehicles are less apt to attack, because the next mech will pound them hard in return. Mechs are also good at capturing contested properties where a smaller tank or recon might look to disrupt you. Not many COs will send in their costly vehicle to attack a Mech sitting on 3 star defensive terrain. Mechs are also useful in stopping a Tank Rush, as they have the same firepower as the smaller tank, but cost less than half, allowing you to save for bigger and better things. Use Mechs instead of Infantry on more forward contested bases, as this removes their key weakness (slow movement), and are generally more useful in attack and defense than the weaker Infantry. Mechs are also useful at destroying the more powerful Md and Neotanks, although it takes several to do so, but they can do it cost effeciently if allowing to Surround and Gang up on the poor singled out units. 
Recons are undervalued by most players. Recons are very useful inside Fog Of War, but no one will debate that. Outside of Fog Of War though, recons are still very useful. Recons are great in delaying capture of enemy infantry, especially if you can shoot them before they get on the cities themselves. Even injuring an infantry or two can delay the enemy from precious funding, and force them to repair those same infantry once they finish capturing (if you don't end up finishing them off). Recons are also useful against Infantry Spamming, but only if they are well supported. A single recon won't turn the tide against a flood of infantry, but a couple recons supported by infantry of your own will make pure infantry floods stop in their tracks. 
APCs are useful in the earlier Expansion Phase, where you can zip ahead and grab extra properties. If you learn how to APC Hop with two infantry, you can quickly transport two infantry to the front lines to use as you see fit. APCs are also useful in blocking key properties from enemy Infantry capture, as they have tank level armor, and do not suffer much damage from machine gun fire. This forces the enemy to use slower mechs to remove your APC, or divert tanks or Anti-airs to do the job. APCs are also useful in resupplying your troops, but unless you rely on Megatanks, Stealths, and Rockets a lot, their resupply issue is pretty moot. Most Units will end up being Crippled to uselessness by the time they need resupplied, or even dead. Unless of course you are battling against Drake. Then the APC will really shine. 
Artillery are very useful units, one of the most cost effective units in the game. Artillery will Cripple just about all of the lower costing units. They also have decent movement and movement type, and can follow up your tanks through heavier terrain more effectively than Rockets. Artillery should mainly be used as a unit to execute Zone Control and as a unit to help damage the larger Tanks. Parking an Artillery behind your tanks and infantry will help deter an enemy attack, as you will come out ahead in effeciency in a battle of equal units. For example, you have a tank protecting your lone artillery against an enemy tank, who doesnt have the movement points to strike at your artillery due to Blocking or terrain. Attacking first would deal 5-6 damage to your tank. Not too shabby, but your artillery would do the same back, plus your own tank would have counter-attacked for around 2 damage, and could then attack to finish off the enemy tank. Using artillery to go on the attack by themselves is generally a bad idea, unless the map has turned into a Stalemate. You should generally use artillery as a means to destroy larger tanks such as Md and Neotanks, freeing up your own to devastate anything in their path, such as enemy artillery, or smaller tanks. 
The first tank is one of the most useful units in the game. They are one of the first units that you can use to exhibit Zone Control, as with their speedy movement, and armor, only another tank can match it, and whoever is in place first will get to strike first. Tanks will deal good damage to enemy artillery that aren't protected well, and against ones that are, inching back is always an option as unless enemy units are Blocking your movement, you have to choose to be in range of indirect fire. Tanks are generally useful in harassing enemy Infantry just like Recons, especially in maps where not many roads are present. Tanks are also useful in destroying AA guns, letting your air units breathe easier as they zip around destroying the enemy. 
AA are designed to make Air Units cringe. A single 8000g AA will be able to OHKO B-copters and T-copters, and Cripple Bombers. AAs also have excellent defense versus B-Copters and have nothing to fear against them. But against Bombers, AAs can be OHKO'd themselves. Luckily, Bombers almost cost 3 times as much, and just having another AA around will deter a Bomber from attacking your AAs. Anti-airs are also useful at mowing down Infantry, and do so very well, much better than tanks and recons can. Anti-Airs are also useful at finishing off weakened Tanks, and will generally deal around 3 damage, enough to be useful as a second strike against a tank that your own has just attacked. They will also do 4-5 damage to an APC, and thus can be used to destroy pesky ones being used to block. However, they should not be built specifically for this purpose.
Contrary to what may be seen elsewhere, AAs are NOT an effective means of dealing with a mech flood. While they do serious damage to mechs on first strike, they themselves are annhialated while defending. At three times the cost, trading AAs at 1:2.66 is prohibitive. 
B-copters are the most common air unit you will see used in games. This is generally due to their low cost, and high usefulness. Treat your B-copters like flying tanks. They generally will do about the same damage as the small tank, yet won't suffer nasty counter-attacks. Since they are a flying unit, their 6 movement is much better than a land based tanks 6 movement. B-copters are nice because they can hover just out of reach of any Anti-Air units the enemy has, and can almost ignore any other unit. B-copters can do decent damage to the larger tanks, and can slowly widdle their hp down. They will suffer damage in return, but since your unit costs 2 or 3 times less than theirs, it's a nice trade off. Use your ground units to smash the enemy's AA capabilities, and your B-copters can attack without fear. Tanks protect your air units from AA, and your air units protect your tanks from other threats. 
Missiles are generally thought of as nearly useless by many players due to AA being so effective, and in many ways, missiles aren't as good. But don't discredit missiles completely. If you don't have to travel through heavy terrain, and can create them near the front lines, missiles are a nice unit to have. Their 5 range will help emit Zone Control for air units, something that doesn't happen to them much. Placing 1 of 2 of these behind your other units will help deter any air attack, and help protect your missiles. A useful tactic is to have AA guns more forward, and missiles protecting them from bomber attack, since bombers will Cripple AA guns. Also, try to wheel up missiles into range of enemy airports, they will help lockdown that airport, especially if you can't reach it due to terrain (like mountains or rivers). Missiles are especially useful in Fog Of War like all indirects, as they can hide in forests and also have 5 vision range. 
Rockets are a very useful indirect unit in that they have 5 range. This is very nice at emiting Zone Control. But Rockets do have their drawbacks. They are wheeled units, which means unlike artillery, they can not travel through the common plains tiles very quickly. Also, they have lower ammo than artillery, cost 2.5 times as much, have lower armor, and have a large blind spot. Rockets are especially useful for countering enemy artillery, but they have to be careful. 2 artillery can move into range of the rocket, and actually win without first strike. Those same artillery can also move 2 to 3 squares away from the rocket, and sit in it's blind spot, while still being able to fire at the rocket. This means you should work especially hard at protecting your rockets, and shouldn't deploy them too early. Rockets are also useful at countering Mega Tanks, as they do 3 damage to them with average luck, and the Mega Tanks are almost too slow to even advance on your Rockets.