Suggestions for Equipping Troops (Apocalypse)
Real-Time
Early game
Always equip a stun grapple as your secondary weapon. Put machineguns, autocannons or whatever else you like as the main weapon and put a stun grapple in the second hand. This serves two functions. Firstly, it gives you a very strong weapon against those annoying close-combat aliens (multi-worms and poppers). Secondly, it will significantly increase your chances of capturing living aliens, something that is essential to developing toxi-guns.
Another idea is have all of your soldiers with two of the same weapon, it will increase your killing power. At first you will have worse accuracy, but it will pay off later as your soldier gets better. Have one soldier who is your 'stunner' with more stun grenades, and dual stun grapples.
Always carry several stun grenades on all your troops. Not only can you use them to capture aliens, stun gas also evaporates faster then smoke, so you can use stun grenades to clear smoke. Incendiary grenades can also be used for this purpose in a kind of rock-paper-scissors-esque fashion.
Lastly, an alien in stun gas will try to get out of the cloud and so it stops shooting at you, which is good. Each troop should always carry a medi kit, as you are bound to get hurt, and medi-kits save lives. Research and equip beam weaponry as soon as you can, since ammo is bound to run low.
Dual autocannons are exceptionally powerful in the hands of the right troopers, especially when loaded with explosive rounds.
Mid-game
As soon as you have toxiguns, carry them on any mission where you will encounter only aliens, and don't carry other guns. Toxiguns have a high firing rate, so you can keep your other hand free for other things (like medkits), and you won't capture many energy shields without toxiguns. This also has the advantage of leaving your troops mostly unable to harm one another if under psionic control (apart from grenades) - toxiguns deal minimal damage to agents, though sustained fire is still lethal. Androids can safely carry any heavy weapons you'd like to bring, since they can't be turned against you.
As soon as you have energy shields researched, equip them. Equip other weapons you would like to use as you research them. The two kinds of alien grenade are quite useful, and some of the other equipment is nice too.
Heavy Launchers and Mini Launchers are effective - if somewhat limited in ammunition and purpose - weapons. Use them for UFO assaults or other situations where collateral damage is not your main concern; the missiles may not be blaster bombs but the do have at least some tracking. Their hefty punch can aid in softening up tightly packed UFO defenders and Megaspawn. You can dual wield these and set them to auto-fire and still hit your opponents, since the tracking supersedes accuracy.
Late-game
Late game tactical combat is rather easy. You should have the energy shields, toxiguns, and Disruptor Armor that you need to take out the enemy easily. For combat against aliens, make sure your troops are equipped with toxiguns, plenty of the best ammo you have for them, some stun grapples to capture those few aliens you still need a live specimen of and stun grenades to take the aliens smoke advantage.
For raiding enemy buildings, you don't need energy shields if you have the Disruptor Armor. Equip your troops with dual Devastator Cannons (if for some reason you want to level the entire structure - a single dev is more than sufficient if you bother to aim), Disruptor Armor and any other weapons you would like to use. Only the heaviest of explosives or concentrated Devastator fire (which is a possibility in the later stages of the game, no matter the raid victim!) has any chance of hurting you with Disruptor Armor, so don't worry much about it.
Teleporters are almost cheating in Real Time, since they work instantaneously while the game is paused. This means you can pause at the precise moment before a projectile hits, or even while an explosive is detonating, and teleport your agent out of the way. Amazingly good.
Turn-Based
Early Game
There's no sense dual wielding in TB, since firing either weapon costs time units, so you don't get "free" shots with your off-hand. You should fire two-handed weapons with nothing in the other hand to increase accuracy; since moving things in your inventory is free you can just keep any secondary weapon in your backpack. This also means you can feel free to bring a stun grapple and equip it as necessary, but be careful of weight.
Auto-cannons are very effective weapons in the early game because their explosive ammo is still quite effective in the hands of agents with poor accuracy. Auto-cannons maintain their effectiveness right through the game against two important enemies - Brainsuckers and Poppers. Auto-cannons are the only weapon that can effectively give reaction fire with explosive rounds, making them very effective at taking out Brainsuckers, poppers and Hyperworms as they charge, since a near-miss will still result in a kill. Your starting armor is very effective, so shooting HE at point blank range won't hurt too much (though switching to AP ammo is free anyway). The downside is that throwing explosives around will cause property damage, destroy artifacts and kill aliens you've stunned.
AP, stun and smoke grenades should be standard issue for everyone. Grenades are great for low accuracy agents to get kills on Hyperworms and Brainsuckers. Stun grenades cause a lot more 1-hit KOs in TB mode, making them superior to AP grenades as a fallback. Drop smoke frequently to protect agents against return fire and to get closer with machine guns and stun grapples.
Equipping some sniper rifles in the hands of your high accuracy soldiers can work well, allowing them to kill enemies at long range. The downside is that their accuracy training is poor and their damage is easily the lowest in the game. Most combat in Apocalypse is short-to-medium range, so it's difficult for snipers to get good vantage points unless they fly (and then they're inaccurate).
Mid Game
First priority is to buy up as much Marsec armor as you can and use it to speed up your low strength troops, especially hybrids, since Megapol armor is very heavy. The chest piece also gives flight, which is a great tactical advantage. Your second priority is to replace your laser sniper rifles with plasma guns; they're not quite as accurate, but they pack more punch, making them much more effective against tougher aliens.
Once you research toxiguns they'll become an important weapon in your arsenal. They have mediocre accuracy, but their fire rate more than makes up for it, and you should replace any machine guns and some auto-cannons with them. Keep some ACs around for their HE ammo capability, preferably on Androids so they can't be used against you through psionics or brainsucking.
Heavy and mini-launchers are useful weapons to pick up. A shot at a location can miss, but a shot at a unit will home and always hit. This makes mini-launchers extremely useful for those shots you can't afford to miss like your last shot of the turn against a Popper, Brainsucker or Hyperworm swarm. Heavy launchers can fulfill the same role, but take more TUs to fire and are extremely bulky and heavy.
Use Proximity Mines and Boomeroids for Brainsucker and Popper defence when you're camping a corridor or UFO entrance.
Late Game
Disruptor armor will neatly solve your weight problems and allow even your lowest strength soldiers to be armed to the teeth. In TB your ability to charge up to a Megaspawn or Psimorph and toxigun it to death without it shooting back makes them considerably easier than in RT, such that even late game your most credible threats remain the lowly Brainsucker and Popper.
Toxigun agents equipped with Teleporters make the rest of the game a joke. Despite the heavy TU cost, agents can teleport to alien spawn points and kill them on turn 1.
Androids and Hybrids
Two opposites, androids are physically strong, while hybrids are physically weak but are the only race that can effectively use psionics. Ideally, androids will have lots of extra equipment to either demolish everything or to replenish teammates supplies and hybrids travel as light as possible but with a Mind Bender even if they haven't received adequate psionic training.
At the start of the game androids will have the best aim of your troops, so should be given the best weapons. With some training and experience your human soldiers will surpass them. Androids' high HP and immunity to brainsuckers and psionics makes them good scouts and shock troops, since turning a corner in to something unpleasant is less likely to result in their death than a human. Also, while by no means expendable, androids are more easily replaced than an experienced human. As such, once you have humans with good accuracy that can take over fire support, equip your androids for short range combat and/or demolition and use them in the front line. Another advantage of androids' immunity to psionics and brainsuckers is that they can safely be given heavy weapons like launchers, since there is no way for the aliens to take control of them.
In the late game once you have disruptor armor and advanced, lighter weaponry hybrids really come in to their own. The lighter equipment makes a hybrid's strength disadvantage less of a problem, and the toughness of disruptor armor and shields makes their lack of HP less of an issue as well. Meanwhile, a hybrid's high psionic defence is an asset even if you don't use offensive psionics much, and their proficiency with weapons is comparable to a human.