A dedicated rain bike is nice. Even an old MTB with street tires works great in the rain and over the winter.

Fenders, as already noted is a must. Even with fenders, shoes are going to get wet. So you'll need shoe covers or dedicated water proof shoes.

Helmet cover, check. I also prefer a helmet with a visor for wet rides. This helps shield some of the rain from the eyes.

Rain pants for heavy cold rain, yes. But sometimes I just ride in shorts if the weather isn't to cold or heavy. Also, I don't like rain pants next to the skin, the pants stick. Instead, use really thin light weight tights, so much more comfortable while wearing rain pants.

Jacket is a must, people have their favorites. My current favorite is made by Endura, also have Showers Pass and a couple others.

Gloves, you'll need something at least water resistant, if not water proof.

Lights are good on days with low light, both front and rear.

Next, is don't be a in a hurry, give plenty of time to where ever your going.