If you decide to go w/ the .177 cal pellet route, I'd recommend a break barrel action gun (only have to pull barrel down and snap back into place once to energize it). Look for one w/ velocity in excess of 1000 FPS, which should provide plenty of zip. They make break barrel .22 cal pellet rifles as well.
If you have a .22 cal rifle, the previous poster's recommend for 22CB ammo is great as well. While the velocity is only btwn 700-800 FPS, the mass of the round is considerably more than a .177 pellet. You could probably knock a boar coon down fairly easy w/ that, let alone a silly rat.
BTW, buying a silencer for a .22 rifle is more trouble than its worth. First, if your barrel isn't threaded, then you'd have to get an adapter, pay big $$$, and register it w/ the ATF and pay them big $$$ to do so. Just go w/ a pellet rifle.