I have been to Three Brothers Liquor Store a few times to quench some thirsts in the past. They carry all the major hard liquors and have a decent variety of rare ones. I think their forte though is beer. The coolers always stocked with Budweiser, Coor, and Miller products. The prices aren't exactly bargains, but if you got a party going on, it doesn't matter, does it?
Cicero's a big street. This is located close to the intersection of Cicero and Augusta Blvd. I wouldn't go there after dark by myself, but it's a relatively safe place. But use common sense whenever you're in doubt.