With a quick search on the internet you will find that you could spend hundreds of dollars on a jacket for bicycling in the rain. They probably work great. You could get Gore-Tex, Supplex or a wide variety of high tech fabrics. Something like this is probably a good idea in situations where exposure or hypothermia is a risk. Realistically we are all mostly riding in warm weather and with a cell phone so you are not going to get hurt if you get wet. All you need is something simple.
For short warm weather rides we each carry cheap old fashioned drug store ponchos. They are great because they are cheap, compact, disposable, and readily available. They are not so good because they are hot and your hands, handlebars and feet get wet. They also tear easily.

A cheap old fashioned drug store ponchos is compact, disposable, and readily available.

A step up from the drugstore variety is the bicycling poncho. We each got one of these from ebay for about $15 in 2012. We have only needed them once but they worked very well. My hands, handlebars and feet all stayed dry. My mirror was still usable. The hood with brim fit fine under my helmet and helped to keep the rain off my glasses. There is a clip to attach the poncho to a brake cable to keep it in place–it works great!

You can get a bicycling poncho for about $15. Works well to keep your hands and gloves and feet and shoes dry. The detachable hood with brim also works well.

It would be nice to have some kind of rain covers for the panniers but that is just another thing to carry. Everything inside the panniers is first put into a plastic bag. It would be a bummer if the panniers got soaked but that has never happened. There is usually someplace to take shelter to wait out a rain shower. We also carry a really nice tarp that can cover the bikes.

Here are our bikes tucked under our cool tarp that our friend Nancy gave to us. If is fabric, has multiple tie downs and packs down very small.