I'm kind of wondering how much better I need to become to 'cut the mustard' in terms of basic army marksmanship.
Are you enlisting? The basic qualification guidelines are as listed by others above. You can find many field manuals online at
http://www.adtdl.army.mil/atdls.htm.
You can get some practice in service rifle shooting through various CMP (Civilian Marksmanship Program)-affiliated Service Rifle matches. Many clubs have "CMP" matches, and High Power or Service Rifle clinics. You can look for a list of events at
www.odcmp.com and you want to get a move on - most New England clubs wrap up for the season in October. Ironically, Massachusetts, that communist, gun-unfriendly state, has more matches, clinics, and clubs than all 5 other New England states combined.
High Power events shoot at 100, 200, 300, and 600 yards. Many clubs shoot reduced range events (using smaller targets) at 100 or 200 yards only. The CMP provides the clubs with loaner AR15's (and Garands) so you don't have to own your own - just show up with $20 or so and they'll supply you with rifle, ammo, targets, and often a spotter or coach.
The Army course is more of a combat course and does not exercise precision aimed fire like known distance courses do. The Marine known distance course requires precision aiming but doesn't exercise target recognition and range estimation skills like combat courses do. Many people from both services think infantryman should do both - but the services may be too hide-bound to try the "Not Invented Here" approaches.
Believe it or not, many shooters miss the 50 meter targets. Here's how. First, put on 32-40 lbs of helmet, body armor, load bearing equipment, water, first aid bandages, 2-4 grenades, and at least half a dozen magazines. Now, walk around for a while until you are sweating and thirsty. Crawl 50 feet or so up a hill. Then lie down on top of all that junk and start firing at ranges very close to what your M16 is battle-ranged for.
Now, without warning, a profile target (a few inches high representing shoulders, and a higher center mass representing a helmet) pops up at close range. Because it is so close, you forget that your rifle shoots a couple of inches high. Bang - right over the target. I used to aim at the "dirt line" and if I skipped one in, who cares as long as the target goes down.
) They popped up from between 50 to 300 meters. At 50 it was just a head and shoulders. At mid-way ranges, it was upper torso. At 300, it was full torso and waist. They were fun to shoot at, but it'd have been more fun if they flew apart or something like that. 
