Here's what I've got:

(lists may be horrendously incomplete and subject to change)

Weapon Damage Types: (symbols on HUD next to weapons in bottom left, or next to attack value in options menu)

• Kinetic (standard, no symbol)
  
• Arc (light-blue, spikey circle)
  
• Solar (orange flame)
  
• Void (purple swirl)

Defense Types: (colours of HP bars when you put your reticle over a target)

• Red - most enemies - (health?)
  
• White - Fallen Captain, Hive Wizard - THIS LAYER RECHARGES - (Shields?)
  
• Yellow - Seen on 'Elder' Fallen, and bosses like Sepiks Prime and the Devil Walker - (armour?)

Inflicted Damage Types: (colours of numbers that pop up beside enemies on hits)

• White - a standard hit, possibly with DMG penalties depending on where and with what you hit
  
• Yellow - A critical hit. Headshots on un-whited-layer bipeds, knees of Devil Walker, 'eye' of Sepiks Prime.
  
• Blue - Arc damage hit on susceptible target (only positive ID vs. shields of Fallen Captain)
  
• Orange - Solar damage on susceptible target (possibly shields of Hive Wizard)

A few ProTips:
Use an Arc damage Machine Gun against the white-layer of Fallen Captains - it cuts them to ribbons. You can either keep using it on their Red (remember to headshot!) or quickly switch to your Primary to conserve ammo.
e.g. I have a Genghis-C Machine Gun, D83, Arc-type - it gets 183 damage per hit on a Fallen Captain's white.
Conversely, using a Genghis-C, D96, Solar-type only gets 48 damage per hit - avoid using Solar weapons against Fallen Bipeds

Solar Fusion Rifles seem good against Hive Wizard Shields. The one at the end of 'The Dark Within', 'Gotra, Eir Spawn', level 6, on Legend, I stripped his shields in one burst with a Solas FR3, D67, Solar-type.

Fallen bipeds melee with their left arms. So if you get it close and step past them to the right (their left), you can get in a cycle where you dodge an attack and can get in a free melee hit (as a Hunter anyway). Good for Fallen Captains if you're low on ammo but, watch out on the approach.

That's all I got (apart from the obvious 'go for the headshots'). Anyone else got any tips?

Use an Arc damage Machine Gun against the white-layer of Fallen Captains - it cuts them to ribbons.

Alternatively, an arc damage fusion rifle seems to work well if you are out of, or want to conserve heavy ammo. The fusion rifle in general just seems great for cutting through shields.

Curious if there are other damage types to correspond to the other enemies.

I found a Breaker SA/5 rocket launcher that deals arc damage, and the description reads "The SA/5 Breaker answers a requirement for a hard-hitting launcher to defeat new varieties of Fallen armor."

Arc - Fallen
Solar (light?) - Hive (darkness?)
????? - Cabal
????? - Vex

Curious if there are other damage types to correspond to the other enemies.

I found a Breaker SA/5 rocket launcher that deals arc damage, and the description reads "The SA/5 Breaker answers a requirement for a hard-hitting launcher to defeat new varieties of Fallen armor."

Arc - Fallen
Solar (light?) - Hive (darkness?)
????? - Cabal
????? - Vex

That does seem to be more-or-less the case. But you don't seem to get any sort of damage bonus except against the shields (or avoiding a penalty).

I think the damage system must not be fully enabled in the Alpha. Criticals don't do as much damage as you might hope and the balancing for Legend on Devils' Lair seems whack for 3 people. Like 15 headshots just to kill the Dregs guarding Sepiks Prime - WTF?

Cody is right about the Fusion Rifle: it kicks ass.

In the Alpha you can carry-over your ammo from the same type of weapon when you change it (e.g. shotgun-shotgun, but not shotgun-sniper). So you can actually switch between different Fusion Rifles for different enemies, keep your ammo, and get the correct damage bonus (or avoid a penalty).

On The Devil's Lair, I brought two Fusion Rifles: one Arc, one Solar. Arc for Fallen Captains and Solar for Hive Wizards. Works a treat. One burst, max two if my aim is off, and their shields are gone.

Also found out that the Hunter's Golden Gun and Grenade abilities seem to do Solar damage while the Warlock's Nova Bomb(?) and grenade do Void. Against the Devil Walker and Sepiks Prime, the Warlock has a big advantage since it's specials don't suffer a damage penalty.

Maybe this was obvious to other people, but I just realized it today :P

Yellow doesn't seem to indicate anything special other than "this enemy has a lot of HP!" and that damage penalties will apply for using the wrong attack type (e.g. Solar against Devil Walker - bad idea) Yellow bars vary somewhere in the order of 10-20 times the HP of the version of the unit with a red bar.

I'm assuming the Beta is going to have all this stuff more finely tuned - I can't wait! This weekend has been fun :)

In the Alpha you can carry-over your ammo from the same type of weapon when you change it (e.g. shotgun-shotgun, but not shotgun-sniper). So you can actually switch between different Fusion Rifles for different enemies, keep your ammo, and get the correct damage bonus (or avoid a penalty).

That's a huge protip right there, and makes the switching mechanic even better. I did not know this.

Ok and I can see how when preparing for the waves part of the strike (where eventually both Captains and Wizards are coming at you at the same time) that you can coordinate damage types with your fireteam having one person responsible for Wizard Shields and 2 people responsible for Captain shields (given the final mini boss of that section).

Then when you move onto the spider have everyone gear up for the spider tank.

The consumables really change my approach to the spider battle. Previously I made sure to clean up supporting enemies for ammo of all types to stay efficient during the battle but once I could afford consumables it turned into a much more straight forward affair, hiding from supporting enemies and dumping heavy ammo into the legs... But I suppose the fun will be figuring out how to make it more efficient.