Go up the speed (Assault Tactics) or accuracy (Marksmanship) lines first, then the Counterinsurgency line. The Carbine Special Abilities line will come as you get the others.
Get some Doc buffs (and Dancer/Musician buffs if you can find them) and head to the Tusken Fort on Tatooine or do Quenker missions on Dantooine.
Optional: Get some clothes with Bio-Engineer enhancements for Stun/Melee Defense. Be aware that the highest that this bonus can accumulate is +25 (even though it will add up higher, it will only count up to the +25).
Gained by killing MOBs with a Carbine ranged weapon.
Use a Laser Carbine, Nym Slug Thrower, or Elite Carbine to level.
Use Leg Shot (hit Action bar), Action Shot (bleed Action bar), Crippling Shot (good damage), Full Auto Single 2 (stun, blind, dizzy, and good damage), Wild Shot (stuns single/cone), Charge Shot (knockdown single/cone), Burst Shot (high damage), and Full Auto Area (stun, blind, dizzy cone).
Carbine Special Abilities:
Combat XP is gained by killing MOBs by any means. It’s 10% of the weapons XP gained.
As you go up the other 3 trees you should have enough Combat XP to advance in this line.
If you’re having problems raising it, remember that it’s a percentage of the XP gained for killing things. Just get some buffs and kill higher end MOBs. For example, Quenkers will give you around 300 Combat XP each!