Cold rain is dangerous. Staying warm in a cold wet situation is difficult. Hypothermia is serious stuff and it can kill you faster than just about any other environmental threat.
In warm weather, I could care if it rains. I get wet but so what? Here in Central Florida in the summer, I get wet from sweat on sunny days. If it rains, at least all I will have is water and dirt on me.
In cold weather, I wear a cheap nylon rain suit I bought from Performance Bicycle for $99 on sale. It is rather cheesy. But I figured that on the bike and in the slop, I am not making a fashion statement. Also whilst on the bike, I am going to sweat almost no matter what. So a fancier fabric is not a big deal. The rain suit works pretty well. I manage to stay comfortable in it. You also should have something to keep your feet dry. I use shoe covers. But I know some people who go low tech. They use plastic bags on their feet. You can also go low tech for your body and buy a really cheap rain jacket or even use a garbage bag.
The garbage bag routine would be an good expedient on a ride that turns wet and cold. You stop at a convenience store and beg for a trash bag and use it to keep dry enough to get home.