Running tips!

Teresa🐘🌿 • Miracle baby born 10/20/18
I recently got a dog and have picked up a habit of running with him in the morning to help him burn some energy. 
I am new to running and need some tips. I've noticed that it is suppose to rain two days this week. Is it okay to run in the rain? Really my main concern is with my new running shoes. I bought an expensive pair of Asics and don't want to ruin them.  
Any tips or suggestions? Do runner typically refrain from running in the rain? Or are running shoes built for things like this?