If you run in the rain you will get wet. There's nothing you can do to stop it happening so you just need to make sure whatever you're wearing is comfortable.
Waterproof jackets are a complete waste of time as you'll get soaked inside from sweat. I stick with technical fabrics, not cotton, and a cap. I wear lightweight shoes so they let water but I have good socks that don't rub when they're wet. If it's cold and windy I sometimes wear a wind proof gillet.
Recommend "tight" trousers as opposed to "flappy" ones. If they are loose and wet through, they flap like cold wet fish against your legs...tis not nice. Running tights (full-length or capri) every time for me...
A cap keeps the rain off your face, which is useful!
The wetter you get, the better you feel once you are back indoors (in theory anyway!)
I just go for lightweight tops and my usual knee length running leggings and get wet. I've tried so many waterproof jackets over the years and they just make me hot and sweaty and uncomfortable. Windproof gilets are good too.
The only time I wear a hat is if it's less than minus 2 degrees otherwise I end up ripping them off my head in frustration after 5 minutes!
I live in the countryside and run off road so my feet always get wet and filthy but I just have a shower as soon as I get back.