I am currently enlisted in the Army.
Basic training instills discipline, introduces stressful environments to get you used to them, and bores some fundamentals (e.g. how to properly fire an M4/M16 or basic movement drills). It basically tries to mold you into a capable, resilient, strong soldier, able to withstand the stresses of the battlefield and overcome your enemy.
As far as sniper school goes, I haven't been. I do know some others that have graduated sniper school. It takes a lot of mental focus more than anything else. Ballistics is a key part of it. They say it's the hardest school, but the most rewarding.