Heya!
Melee and Ranged combat are handled the same way. The only real difference is the bonuses applied to the rolls to hit/strike. For the purposes of expediency, I'll illustrate an example where someone brings a knife to a gun fight.
The important things to know for combat are the skills and attributes involved, and the number of Action Points each participant has.
So, as a quick run-down, for melee combat, your Agility bonus adds to your rolls to strike/hit, and your Strength bonus adds to your damage. For ranged combat, your Aptitude bonus adds to your rolls to hit/strike, and your Intelligence bonus adds to your damage.
The above bonuses will be in addition to the rank of the appropriate skill (blade for knife strikes, pistol for handgun strikes, etc.), and whatever other bonuses the characters may receive from equipment, augmentations, other skills, etc.
All characters begin with 3 Action Points, and can gain more based upon their highest attribute pool bonus, and a couple of augmentations. Base Initiative bonus is also based upon the highest attribute pool bonus as well.
So, lets take a couple of folks, we'll call them Carrie and Will. Carrie has a high-frequency sword and Will has a standard 10mm auto-pistol.
If we assume they both have 4 Action Points and have managed their attributes to both have a +1 bonus to their relevant attributes (Strength and Agility for Carrie, Intelligence and Aptitude for Will). Let's also assume that Carrie has the Combat Proficiency: Blade skill at Rank 2, and Will has the Combat Proficiency: Pistol skill at Rank 2.
All of that distills down to the below:
- Carrie has 4 Action Points, and a bonus of +3 to strike with her sword (+1 from her Agility, and +2 from her blade skill).
- Will has 4 Action Points, and a bonus of +3 to strike with his pistol (+1 from his Aptitude, and +2 from his pistol skill).
- They both have a +1 to Initiative, and a +1 to damage with their chosen weapons.
- Let's assume they start about 15 feet/3 squares apart, are wearing armor with 5 Soak, and have their chosen weapons in hand.
To begin, we roll for Initiative. Both players roll 2D10 and add their Initiative bonuses:
Carrie rolls an 8 and adds +1 for a total of 9.
Will rolls an 11 and adds +1 for a total of 12, so he'll go first.
Will declares he's going to shoot Carrie with a single round, which is a Short Attack Action, so he spends one Action Point and rolls to strike. His 2D10 roll totals 12 and he adds his bonus of +3 for a total of 15.
Carrie may now decide what she wants to do in response. She can try to Dodge, which will cost 2 Action Points and avoid all damage if she succeeds, or she can try and Guard, which will cost 1 Action point and reduce damage by half if she succeeds. She can also try to Counterattack, which will cost 2 Action Points and, if it succeeds, means they will both take damage from one-another's attacks. That last option isn't great for Carrie, though, since her sword isn't going to reach 15 feet/3 squares, unless she throws it, and she isn't ready for that kind of desperate action yet.
Alternatively, she can always just choose to do nothing, to take the hit and save her Action Points.
Wanting to save her Action Points to close and attack, Carrie decides to take the hit, so Will announces that the total is 11 Physical Damage, 10 for his gun's base damage, +1 for his Intelligence bonus.
Carrie subtracts her armor Soak of 5 from the 11 damage, reducing it to 6, then applies that to her physical Health, she's now taken 6 points of Physical Damage.
Will now has a decision to make, since he's made only one short action this turn, he can make another, or he can hold fast and save his Action Points for later attack and defense. A short action is any action that costs only 1 Action Point, so it could include a movement, another short attack, the swapping of weapons, using a skill, etc. In this case, he decides to save his Action Points for later and ends his active turn.
Now it's Carrie's active turn, and she needs to close on Will so she can bring her sword to bear. She declares a movement, spends 1 Action Point, and advances on her opponent. She can move a number of squares equal to half her Agility attribute, which would be at least 8 in this case, since her Agility is high enough to give her a +1 to melee strikes, which means it's at least a 16. So, she advances the 3 squares to bring herself into melee range with Will, then declares a second short action, this time a melee attack with her sword.
She spends another Action Point for her attack (now having spent 2 total, one for the move and one for the attack) and rolls 2D10 to strike. She gets double 6s on the dice, which is a critical success, so she has a decision to make.
Either she can take the critical success, which means she automatically hits, unless WIll declares a Guard, Dodge, or Counterattack action and then rolls doubles 6 or higher (double 6s, double 7s, etc.), which is possible, but the odds are against him; OR she can take a point of Adrenalin and treat her roll as a normal roll, adding her bonus as usual. The Adrenalin gives her an extra Action Point that she must spend in this Round of combat.
Since she's decided to go full offense for this fight, Carrie takes the point of Adrenalin, giving her a bonus Action Point, then adds her normal +3 to strike to her roll, giving her a total of 15 to strike.
Will decides to try and Guard against the attack to hopefully reduce the damage he'll take, so he spends 1 Action Point and rolls the dice. He gets an 11 and adds his bonus to Guard (by default this is either the character's Aptitude bonus, or their Agility bonus, whichever is higher, plus their rank in the Combat Proficiency: Defense skill, if they have it). Since we haven't established that either of these characters has the Defense skill, Will only adds his Aptitude bonus to his roll, for a total of 12, which is lower than Carrie's strike roll, so he takes full damage from her slash.
Carrie informs him that her high-frequency sword does 15 Physical Damage, plus her Strength bonus of +1 for a total of 16. WIll subtracts his armor Soak from that number, making it 11, and applies the damage to his character.
At this point, Carrie's active turn is over, since she's performed two different short actions. As it stands now, she's taken 6 Physical Damage and has 3 Action Points remaining (she used 2, but received a bonus of +1 for the point of adrenalin she took).
WIll has taken 11 Physical Damage and has 2 Action Points remaining (because he shot once and tried to guard once).
It is now Will's active turn again and he decides to end this, so he declares a short attack action, firing his pistol point blank into Carrie's abdomen. He spends 1 Action Point and rolls a 13. This time he adds his normal bonus of +3, and an additional +3 for firing a pistol at point blank range, and gets a total of 19 to strike!
Carrie decides to Counterattack with her sword, so she spends 2 Action Points, leaving them both with 1 Action Point each, and rolls her attack. At this point, she only needs to roll a positive number that is not a critical failure (double 5s or lower by default). She rolls and gets a terrible 3, but it isn't doubles and it's higher than zero, even before she adds her +3 to strike with her sword (note that she doesn't get the point blank bonus to strike with a melee weapon, since that's the only place they work anyway), so her Counterattack is a success.
This time, Will's gun will do 16 Physical Damage (the normal 10, +1 for his Intelligence bonus, and +5 for the point blank damage bonus), and Carrie's sword still does 16 Physical Damage as well. They each subtract their armor Soak of 5 and apply 11 more damage to their character sheets, so WIll has now taken 22 Physical damage, and Carrie has taken 17.
Will's not feeling so good, and he decides to go all in and try to finish the fight, so he declares a second short attack action, spends his last Action Point, and rolls to strike. He rolls double 1s on the dice! This is bad. The GM determines that Will fumbles his gun in the heat of the moment, dropping it to the ground and leaving himself unarmed, and out of Action Points!
Carrie, deciding to take the opportunity to gloat a bit, laughs haughtily at Will (an Open Action, so it costs her no Action Points), then declares a short action to kick WIll's gun away from him so she can finish him off without getting shot at in the next Round. She spends her last Action Point and the Gm declares her kick to be a hand to hand strike, but Carries doesn't have Combat Proficiency: Hand to Hand Combat, so she'll make her roll with only her Agility bonus (since Hand to Hand is still melee combat, her Agility bonus applies). She rolls a 10 on the dice and adds her +1 Agility bonus, which is a hit (because its positive and not a critical failure), kicking WIll's gun well out of reach.
Now, since both WIll and Carrie are out of Action Points, the Round is over. Both characters refresh their Action Points back to full, everyone rolls for Initiative, and the next Round begins.
So that is the basics of combat (both melee and ranged). I tried to keep it pretty basic, but did add a couple other considerations in there, such as critical success/failure, so hopefully that helps out a bit.