Ranged Init, Melee Init and Physical Init all work the same way. Player experimentation has shown that for every 600 skill in the init, attack speed is reduced by 1 second and recharge speed by 2 seconds. People refer to the overall speed of a weapon by separating attack time and recharge time with a "/". The fastest a weapon can act (its "capped" speed) is 1/1.
Incidentally, the base speed of fists when using the Martial Arts skill is 1.25/1.25. Ironically, however, as someone practicing Martial Arts in AO gets better, their fists slow down. The slowing is to the tune of 0.1s at 500 Martial Arts skill and 0.2s at 1000. That makes fists at 1000 MA a 1.45/1.45 weapon.
Weapon: OT Kerans Automatic Grinner
Base speed: 1.5/1.45
To get this gun to 1/1 at full agg, first subtract the .25/.25 bonus of being full agg, making it 1.25/1.20. Since attack is higher than recharge and gets less benefit from init skill, calculate how much skill is needed to drop attack by .25: 600 * .25 = 150. This would make recharge drop by .5 (double the attack benefit) to .7, but it gets capped at 1.
The benefit to raising your skill further than where you are 1/1 is that you reduce your aggdef slider to take more advantage of your duck, dodge, and evade skills, reduce critical hits on you, make you less susceptible to init debuffs, and let you equip a scope to increase your critical hit chance while mitigating the scope's init penalties.
Note that for inits over 1200, the effectiveness of each point above 1200 is drastically reduced. Chronita has figured out that each point over 1200 is worth only one third of a point up to 1200. 1200 would cap a 3.25s attack speed and 5.25s recharge speed. For weapons with attack or recharge greater than that you'd need three times as much initiatives to reduce their speed by a given amount of time; 1500 inits would drop attack another 1/6th of a second and recharge another 1/3rd of a second.
Nano Init works differently. For every 200 skill, nano attack speed is reduced by 1 second. Unlike weapons, nano attack speed does not cap at 1 second, so if you can get it to 0s, you "instacast". Also unlike weapons, recharge speed is never affected.
Thanks to the AO Soldiers Basic Initiative Guide, I learned that the neutral point for nano inits is 50% aggdef instead of 87.5%. That would make full agg benefit 1 second and full def penalty 1 second. This is consistent with my own experimental results and resolves the ambiguity that you may remember reading here in the past. Some player experimentation has suggested that the aggdef bonus/penalty modifications to nano execution speed only happens at full agg or full def; ie, that 1% to 99% is all the same.
I'd been planning on putting a calculator on this page to help with figuring various numbers related to agg/def and initiatives, but Novaprime has made one on the Unity of the Rose web site so now I don't need to. Just visit the AGG/DEF Calculator, where you can calculate inits from weapon speeds, calculate agg-def position for best defences at full speed, and calculate martial arts attack speed from MA skill.
The basis for most of the basic information here was the fine work done by Chronita, one of the Professionals for the Martial Artists, published in the Phys Init/Defensive Skill FAQ in the Martial Artist forum at the official Funcom boards.