I rode my touring bike-actually a converted mountain bike-in the rain a few times. I didn't have fenders, but they are a good idea. I think you can get relatively inexpensive plastic ones for both front and back.

I had racks with panniers and put my stuff in plastic bags. Other than that, I wore a Goretex running suit as rain gear. This was an early generation fabric, so it wasn't all that great, but it did keep me mostly dry. I also had leg and arm warmers. You can get shoe covers, which I didn't have, to keep your feet dry.

Just keep in mind roads can be quite slick, even after the rain stops. I crashed once when someone pulled out in front of me and I had to jump on the brakes to avoid them. I wasn't touring, just riding on the street. I wound up with a big patch of road rash on my thigh that took about two weeks to heal.