Most of the Palladium books I have seen (I'm looking specifically at Macross II - the Role Playing Game) have a section "Determining the number of attacks per melee and Combat Bonuses" in the Combat Rules section:
"Players will find two skills that exclusively determine the pilot's number of attacks with piloting a high-tech robot or power armour: 1) The pilot's normal hand to hand combat/fighting skill and 2) the pilot's Robot Combat skill. [...] The total number [of these 2] indicates the total number of attacks possible."
The same is done for bonuses to strike, parry, dodge. This means if the pilot has +2 to dodge in hand to hand combat and +1 to dodge in robot combat, he has a +3 to dodge when piloting the robot.
So lets say for the sake of example, you have a character that has 3 attacks in regular h2h combat + some robot piloting training that gives him another 3 attacks. When he is piloting a combat mech that he is trained for, he has a total of 6 attacks with stacking strike/parry/dodge bonuses.
If that robot has 8 weapons, that pilot with 6 total actions could pick the same weapon and fire it (as Strikes) 6 times during a combat round. Or he could fire 6 different weapons. (he'd be forfeiting all of his dodges though against incoming Strikes if he did this though - dodges consume an attack action). Some weapons might be restricted to being useable only once or twice per combat round (like the twin cannons on the heads of robotech Valkyrie) . Note that as he is doing this, in-between each shot, opponents are reacting in order of init (spending their actions returning fire or dodging).
Some weapons can be fired as volleys or bursts. If one of his weapons is a 6 rack missile launcher, he can use 1 attack to fire all 6 missles. Or just 1 or 2 or 4 ect.
Some weapons can be fired as bursts - you get +1 to strike and deal a multiplier on the damage (x2, x5, x10! for short, long and 'entire magazine') consuming 20%, 50% and 100% of the magazine ammo. Firing the entire magazine = 2 actions taken at once. But a regular 'single' shot, short or long burst takes only 1 action) Details on if this can be done with a particular weapon are listed in the weapon description.
Someone with a lot of attacks might be able to have a lot of free 'undodgable' attacks after everyone else has no more attacks left. For example if I have 5 attacks and you have 4, and I win init. We start resolving attacks in order of Init. I use my 'attack action' to Strike and each time I do, you use your 'attack action' to Dodge. I still have my 5th attack at the end of it, you've been too busy dodging to return fire and you can't dodge that last Strike.
If I had another ally (or gunner) with me who also had five attacks, we could be Striking twice each time in order of our init. You could do Dodge,Dodge in response the first time. Then Dodge,Dodge in response the 2nd time. But then you are out of actions completely, and going to be taking some serious hits from our 3rd, 4th and 5th attacks spent as Strikes.
Attacks are made in Init order, and Init is rolled at the start of combat and re-rolled at the top of each round once all characters have used all their attacks.
You may be able to find other specific answers or guidance at http://palladium-megaverse.com/forums/
Basically though - its the same as normal combat outside of mecha except that:
Additional attacks and bonuses from Robot Combat are stacking with H2H combat bonuses.
Inside the mech, ALL the weapons are effectively 'readied' at all times (once he is in the mechs its as though you had a fantasy RPG character that had a sword and a mace and a bow and a bottle of alchemists fire and 4 attacks per round but all the weapons readied at once - he could hit someone with the sword 4 times or lob 4 bottles or 1 of each etc.) Some mecha weapons may require spending an attack action to reload them or be otherwise limited in their Rate of Fire description.