Advice! Packing for Spain Late January - Late May

I will be traveling to Spain in two weeks, and I'll stay for about 4 months. Although I'll arrive in Madrid and spend a few days, I'll be stationed in Andalusia for most of the time. Because I am from a place that doesn't really have cold weather, I am in doubt about winter attire. Should I consider a long wool overcoat, or is that an overkill? Is a waterproof trench coat a better idea? I will pack a puffer jacket, boots, gloves and scarfs