You don't get to choose anything. Its the Army, they TELL you what to do. I don't wanna be a SAW gunner, but when we go on dismount stuff I'm packing the SAW. As a Private, you will most likely get the SAW, like it or not, after all, it is an "E-3 slot". When you move up the food chain, you'll hand it off to some other dumb private.
If you are a good enough shot, or you get elected to go, you will attend a Designated Marksman school(a mini sniper school if you will). You could go to the class and not neccessarily be a Designated Marksman. My buddy went to the class, he has an M4 with an ACOG on it. Don't think you get fancy stuff just becuase you've attended a class. If your unit has a weapon that they think is suitable for you(Mk14 or M14, or whatever the hell they have) then they might give it to you.