Hi!
Here's how I do it: Start slow, with some things that she is used to - like brushes or washcloths.
I would start working all over her body with something she is used to, like during a grooming session, and then do the same thing with a small towel. Show her the object, let her sniff it, touch her with it - neck, shoulder, etc. - until she looks comfortable with the concept. Then gently rub the towel over her back, neck, cover her head with it, and then rub it down her leg as far as she'll let you. If she's not spooked over that, do the same thing with a larger towel, and then lay it across her back like a blanket. You might try using a soft rope or lead, and again do the same thing, then drape the lead over her back like the straps for the blanket would be, and pretend to tie it under her belly, so she gets used to that as well.
Put the towel on, take it off, like you would a blanket, until she's totally bored with the procedure. With a baby, you may not get through the whole process in a single session, maybe more like a week of daily (short!) sessions. If she starts getting scared or cranky, go back a step, give a quick brush with her usual brush for example, and then quit for the day. Then start it all over the next day.
Then, once you buy the actual coat, get her accustomed to it, as above, and she should not fuss (unless she's warm and doesn't want her coat on!). My horses see the blankets coming and stick their heads right through the neck opening for me and wait for me to fasten the buckles. These little guys catch on quickly that coats are comfy!
Oh, and just fyi: My gelding will undo velcro on himself and anyone around, so you may want to find a coat with buckles if your filly is the investigative type. Sometimes you can duct-tape the velcro straps slow them down, but that doesn't always work for long...
If you're looking for shopping suggestions, I bought my winter horse coats from KVVet (
http://www.kvvet.com); they are reasonably priced and work really well, and they come in 3 sizes. My 32" mare wears the medium and my 36" gelding wears the large. Look for the Polar Shield Waterproof Foal Turnout Blanket.
Sorry if this is too much info... I get carried away, sometimes...