The real answer is that there isn't one service process but that the service design process depends on who does the work, for who, in which context and for what goal. I've already collected rough notes about 30+ processes that could be called "Service Design Processes".  

But if I had to choose one process to share as a starting point I think that the double diamond model is a good basis that many Service Designers use as an inspiration.

Finally if I had to share a representation that shows how a Service Design project feels I think the Design Squiggle is perfect.

Check out all the questions about the basics of Service Design.