"looking forward to getting started" is correct and usable in written English. You can use it when you are expressing excitement or enthusiasm about beginning a new task or project. For example, "We're all looking forward to getting started with the project this Thursday.".
I'm looking forward to getting started.
I'm so looking forward to getting started".
"We're just looking forward to getting started in the playoffs".
"I enjoy tough conditions and am looking forward to getting started".
"I look forward to getting started immediately and working closely with the board on the business review to ensure that we are able to deliver strength and value to all of our stakeholders and re-emerge as an attractive, solidly performing group".
"It's an exciting opportunity and I look forward to getting started".
"I was really looking forward to getting this year's tournament started," Stosur said after the match.
Ludwig does not simply clarify my doubts with English writing, it enlightens my writing with new possibilities
Simone Ivan Conte
Software Engineer at Adobe, UK