All of the weapon skills -- Handgun, SMG, Assult Rifle, Shotgun, and Sniper Rifle -- decrease recoil and reload time for the appropriate weapon, but will cost three of the four available slots to use weapon mastery. There is a slight advantage in having weapon mastery, but it's not useful enough for most players, unless you plan on sticking to a single weapon type.
Choosing a level one weapon skill would be sufficient, with the remaining three slots better used for support skills.
Hawkeye is pretty much useless, since first-person mode is a handicap in a way, and extra zoom wouldn't improve the weapon's range anyway. Skip it.
Surveyor is only useful if you rely on lock-on, which isn't recommended, since you can't get head shots. Those who do use lock-on find it most helpful in close quarters since it's harder to track enemies, which makes extended range pointless. Not a good choice.
Quarterback lets you throw farther, which can be useful in a grenade throwing match, but not a needed skill. It's more useful if you plan on using Electronic Locators. As a plus, it only takes up one slot even at the highest level.
Trickster lowers the time it takes to set up traps. Again, useful if you plan on playing trap-heavy, but not as versatile as other support skills. Also only takes one skill slot.
The CQC skill lets you grab enemies and perform more hand-to-hand skills, including throat slitting when paired with the Blades skills. Both skills are slot intensive, and require a commitment to a close-quarters play style.
Runner is a useful skill to have leveled-up and at the ready, as it can be very helpful in Base and Capture modes, but less so in other game modes. After leveling to full, keeping a skill slot filled with level one runner will keep you a bit more mobile than without it.
Box move let's you move faster while inside hiding inside a cardboard box. At level three it also has the ability to ram opponents with the box. If it didn't take up to three skill slots it might be fun to have, but as it is -- it's just not worth it if you want to play seriously.