Just want to say that I'm not going to be covering all the books in the series in this level of detail, but this being the core rules I feel it needs a more detailed overlook.
Like most modern RPG Mongoose Traveller has an "roll for a initiative" and subdivided rounds.
Initiative is a DEX check, in that it's 2d6 + DEX DM. High scores goes first. Initiative rolls can be modified by leadership or tactics skills.
Each round for a character is 1 minor action and one significant action. Movement is a minor action and gives you 6 meters (4 squares). You can convert your significant action to two minor actions. You get as many free actions as you can talk the GM into, as per normal in modern RPG :-) A character may also "react" to other characters actions, allowing them to dodge or parry an attack. You may take as many reactions as you want, however each one reduces your initiative by 2 and gives you a minus one on all skill checks and attacks.
Attacks are significant actions. Melee attacks are based on STR or DEX plus your melee skill level, shooting attacks are based on DEX, plus either your gun combat or heavy weapons skill. Thrown weapons are based on DEX, plus your athletic (co-ordination) levels. Each weapon class has a modifier based on the range to the target. Human scale ranges go from "Personal" (under 1.5 meters) to "Distant" (Over 500 meters)
Unlike Classic Traveller, the type of armor you are wearing doesn't effect your chance of being hit, it removes some of the damage done to you. Armor ranges from a leather jacket ("Jack") at one point, to Battle Dress at 18 points. Mongoose Traveller doesn't have the characteristics based modifiers for each weapon, however it does have either "recoil" or "heft", which is a number that you must beat with your STR DM, or suffer a modifier to your initiative the next round equal to the recoil minus your STR DM. Damage done is based on the weapon plus the effect of the attack roll. A attack that has an effect of 6+ always inflicts at least 1 point of damage, regardless of armor worn. Damage is applied to END, then STR or DEX. A character with 2 characteristics at 0 is unconscious and is dead if all 3 are 0.
Automatic weapons have 3 options, single shot, burst or full auto. Burst fire adds the "Auto" rating to the damage done. In full auto you roll a number of dice equal to the auto rating and then group them any way you want. The resultant attacks can be applied to any set of targets within 6 meters of each other. You also only get skill level 1 with a full auto attack.
Vehicle combat is a bastard mix of the space combat system with human level damage. To put it mildly it's not very well done in the basic rules Once the "Vehicle Handbook" (Supplement 5/6) is added in it becomes much better. There is another version of vehicle combat in "Hammers Slammers", which really isn't compatible with the Book 0 + Supplement 5/6 system. Use one or the other. My personal preference is the Supplement 5/6 system.
I'm not fond of damage being applied to your characteristics, but it's an integral part of the Traveller system. I'm still working out how I'm going to replace it, if at all. The optional "knockout" and "random first blood" rules make it some what better, but I'm still not 100% happy with the mechanic
Next up, Encounters, Dangers, Healing and Patrons