You have received some great advice above. Layers are key. Nothing should be made of cotton, all synthetic or merino wool, your choice and budget. Your local sports store is the best place to start looking for this type of clothing, not a fashion outlet. That time of year, you will get a bit of everything. Have a t-shirt for those few warm days near the end, its your base layer early on. Have two long sleeve shirts, a few days you will want them both on at the same time for warmth. Take a fleece as well, the zip up "hoodie" type, not the pull over type. Have a wind layer, either a rain jacket, or a thin nylon one if you have a poncho as well. There will be some windy days. On the cold, windy, frosty morning, you might be wearing everything in your pack when you set out, stripping them off as the sun come out. Later in the camino, its all in your pack, so make sure everything is light weight.
Don't worry about this too much, except for the rain gear which you should practice with before you set off. Everything else can be picked up (or discarded!) along the way if you find the weather turns out to be colder or warmer than the climate graphs suggest.
Buen Camino!