There's no need to select any of the challenges in order to start them - all the challenges are always being tracked. The only caveat with this is that if you start using something you haven't unlocked, you won't get credit for using it.
This applies to things like weapons in the default loadouts before you've unlocked those weapons, or pointstreak rewards for care packages when you haven't unlocked the relevant pointstreak yourself.
A disabled challenge is grayed out in the challenges menu in the barracks if you can't earn points towards it. If you hover over a challenge and down the right hand column of the screen there's a square and/or a rectangle with a "lock" icon over it, that typically means that completing that challenge unlocks an emblem (in the case of the square) or a title (in the case of a rectangle).
If you're using a weapon you've unlocked by gaining levels, and you're not seeing progress, the only other thing I can think of is that you might be playing on dedicated servers on the PC. Infinity Ward allowed dedicated server support on the PC with Modern Warfare 3, but you won't gain XP, weapon XP, progress towards challenges, etc if you play on these servers.