Short answer: you won't stay dry. Your best option is to maintain comfortable body temperature.
Longer answer: What to wear depends how cold it is, how hard it's raining, and whether your bike is equipped with fenders. In the Pacific Northwe't, we have a lot of light rain and a moderate temperature band. I keep the fenders on year-round (which makes me stick out a lot when I attend events in eastern Oregon ;-) and assume that I'll get wet one way or another. How I dress depends on the anticipated temperature band:
Above 70 F (21 °C): dress as I would if it were dry, namely wicky polyester jersey & socks, spandex shorts. These things air out pretty quickly. I'm usually working hard enough that staying warm is not a problem.
50 - 70 F (10 - 21 °C): A Merino (non-itchy) wool jersey works very well in this band. It holds water, but it'll keep you warm. If it's especially windy or approaching the lower temperature range, I'll add a layer, wear wool/poly socks, a beanie, and a polyester light fleece set of tights. The main purpose of these is to stay warm, especially if I stop (e.g., to change a flat).
Below 50 F (10 °C), it's cool enough that I'll definitely be wearing tights and a rain jacket. I'll also have the closed-finger gloves. For my feet, I'll cover the shoes with a neoprene bootie.
Shoes: The neoprene booties don't ward off water as much as you'd think they would, but they do keep my feet warm. They go on sale for ~$20 at various places. I've been very tempted to wear sandals and wool socks.
Leg wear - the polyester fleece tights tend to go on sale at Costco for ~$20 in the September timeframe. They're great for this temperature range.
Rain jacket: I have a nice Showers Pass jacket with lots of zippered vents and pockets. I find that I tend to keep the underarm zippers open to regulate heat except when it's really cold. When I get warm enough, I'll roll it up and stuff it in the handlebar bag.
Head beanie: is a polyester, light fleece cap that I picked up at REI (though many other places have them).