Diablo – What attack does the Wizard do when attacking with a spell while out of Arcane Power

According to Bashiok:

...faster speed weapons consume more resource but may do more damage. Channeled skills might still be better with slower weapons though because on a channeled skill, you have as much mobility as you want, so the "mobility" and "overkill" advantages of faster weapons are eroded with channeled abilities, so all you’re left with is increased mana cost for increased damage.

So yes, a faster weapon will cause more skills to tick faster, resulting in you dealing more damage over time, but will result in you using up more resources as you are channeling faster.

Channeling mechanics

Let's use Ray of Frost as an example,

Cost: 20 Arcane Power

Effect: Channel a beam of frost that deals 215% weapon damage as Cold.

Since this is a channeled skill. The resource cost ticks by 10 and drains that much Arcane Power at twice the speed of your weapon. If your attack speed is 1 attack per second, instead of draining 20 Arcane Power once per second, it will drain 10 Arcane Power every half seconds.

Additionally, Since Ray of Frost is a channeled spell. It is broken down into many small attacks that deal a total of 215% weapon damage based on your attack speed. If your attack speed is 1 attack per second, it will deal a total of 215% weapon damage per second.

tl;dr: The faster the weapon attack speed, the faster the drain; slower the weapon attack speed, slower the drain.

Damage-dealing

Assuming two weapons have the same damage, the faster weapon will use up resources faster than the slower one when channeling, but the faster weapon will take less time to do full damage than the slower one.

• An axe at 15 damage and 1.0 attack speed will drain slower. (15 DPS)

• A wand at 15 damage and 1.5 attack speed will drain faster, but will do
  the same damage faster. (22.5 DPS)

Here's the comprehensive list based of Blizzard's skill calculator and this list of affixes. No base skills increase Arcane Power regeneration, but many skills have a rune that does so. I've only listed those affecting regeneration, as per your question. Many more runes, etc, reduce the Arcane Power cost of the skills or increase the maximum Arcane Power the Wizard can have:

1. Active skills:

   • Magic Missile (Attunement) - Whenever Magic Missile hits a target you gain 4 Arcane Power.
   • Shock Pulse (Lightning Affinity) - Every target hit by a pulse restores 2 Arcane Power.
   • Spectral Blade (Siphoning Blade) - Every enemy hit grants 1 Arcane Power.
   • Electrocute (Surge of Power) - Gain 1 Arcane Power for every enemy hit by Electrocute.
   • Arcane Torrent (Power Stone) - Every missile hit has a 2% chance to leave behind a Power Stone that grants Arcane Power when picked up.
   • Magic Weapon (Conduit) - Attacks have a chance to restore 1 Arcane Power.
   • Familiar (Arcanot) - While the Familiar is active, you regenerate 2 Arcane Power per second.
   • Energy Armor (Absorption) - You have a chance to gain 4 Arcane Power whenever you are hit by a ranged or melee attack.

    

2. Passive skills:

   • Power Hungry - Gain 30 Arcane Power whenever you are healed by a health globe.
   • Astral Presence - Increases your maximum Arcane Power by 20 and Arcane Power regeneration by 2 per second.
   • Prodigy - When you deal damage with a Signature Spell (Magic Missile, Shock Pulse, Spectral Blade, Electrocute), you gain 4 Arcane Power.

    

3. Item affixes:

   • Foreboding / Ominous on Orb or Wand or Wizard Hat - +Arcane Power on Crit