I read the title of your thread and I thought: clothes?!

I pretty much only run in the rain. I run in the winter season and, as I live in the same corner of the world as you do, it means running in the rain or showers or drizzle at least half the time. When it stops raining, I hop back on my bike.

10 Celsius or 50 F, I will wear 3/4 pants (not full tights, too warm for me), a longsleeve, light wool shirt and a hat. (I always wear a hat.) Socks don't matter much unless I'm going to be running for more than half an hour (go for wool) or it's cooler than 10C (go for wool).

Between 5 and 10C (40 to 50F), I add a vest, sometimes light tights, and I wear a warmer hat (but still baseball-cap style). If it's closer to 5 than 10 I also put on a light pair of gloves, which I often take off after a few minutes, but I can use them to wipe my nose.

Between 2 and 5C (34 and 40F), I will go for light tights, and a light jacket which may or may not be a rain jacket. Add a wool hat that covers my ears.

Around freezing point, which doesn't happen very often, I'll might add plastic bags around my feet if it's really raining and I'm going for more than 20-30 minutes, and wear thicker tights, the kind I'd wear for cycling at the above-mentioned temperatures. I did a 2-hour run in a combination of rain, snow, and sleet, and eventually my feet got wet but it was a riot nevertheless.

The trick is to put your shoes on and kick yourself out of the warm house. After a minute or two, it feels fantastic.

Have fun!