1998 Mazda 626 1998 maza squealing/screeching sometimes upo

I intend to give my 98 mazda to my son who will be receiving his license within a couple of months. The car drives well and I haven't had any major problems. Within the last several weeks, I've noticed that sometimes upon start up, after turning the key the car makes a very loud squeal and/or screeching sound and will do this for about 10 seconds or so. I tried to determine if it was happening only under certain conditions (colder days); however, it has occurred on both warm and cold days (more so on colder days though). The longest time it has done this has been for almost one full minute (it seemed like forever). I'm wondering what could cause this and does it require a mechanic or is this a "do it yourself" fix. My check engine light is not on. It only occurs upon start up, not while driving, and not when turning the car off.