Kill MOBs around starting cities (Mos Eisley is great) for levels 1-3 for all the weapon trees.
For level 4, 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. Another great place is the Singing Mountain Clan Cave on Dathomir, just kill the quick-spawn Slaves in the bottom to level quickly with little danger.
Overcharge Shot 1 is powerful, but also really slow and will rapidly decay a weapon if repeatedly used.
Speed and damage Power-ups are your friends, especially at the lower levels. Be aware that they will also increase the decay on your weapons.
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 Rifle ranged weapon.
Use a DLT20 to get level 1, a DLT20a until level 2, a Laser Rifle until Level 3, and then either the Laser Rifle or a Spraystick for the rest.
Use just regular shots or Head Shot 1 then 2 (hit Mind bar).
Gained by killing MOBs with a Pistol ranged weapon.
Use a D18 to get level 1, a DL44 until level 2, then a Scout Blaster for the rest.
Use just regular shots or Body Shot 1 then 2 (hit Health bar).
Gained by killing MOBs with a Carbine ranged weapon.
Use a DH17 to get level 1, a DH17 Short until level 2, an E11 Carbine until Level 3, and then a Laser Carbine for the rest.
Use just regular shots or Leg Shot 1 then 2 (hit Action bar).
Ranged Weapon Support:
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!