As a preschool teacher and mom who has a preschooler, here's my suggestion:
Raincoat (if they play in the rain)
Rainpants (if they'll help kiddo put them on)
Fleece or warm sweatshirt for underneath raincoat
mittens on a string, and pin into coat at tag (make sure string/ribbon is long enough, but mittens aren't dragging)
Warm hat
Boots
I would mark all of these items with your child's name and leave them at school for use there.
You will have colder days which will require warmer boots/down-type jackets, and you may have to bring those 'the day of', but having a little cache of stuff at school will likely help. At my preschool, kids are to be dressed for the weather daily, and I keep a stash of gloves which are washed daily after use, because they get wet and dirty. (I run my program out of my home, and space for coats/boots is limited to while the children are present--I don't have a lot of extra storage--, however, when I move my business, the cubbies will be stocked!)