Would it be OK to chill my wort in my kitchen sink with ice constantly being placed next to the fermenter? My first batch will be on Wednesday and I don't want to screw it up.
Yes, that will work, especially if you're doing a partial boil. The less wort you try to chill, the more effective an ice/water bath is. It also helps if you stir gently with a sanitized spoon. The cooler wort along the pot touching the ice bath will be cooler than the wort in the middle, and stirring gently will speed up the heat transfer. Stir the ice batch once in a while, too.