The phrase "get started" simply means "start".

You use "get started on ___" to talk about work that needs to be done:

Do you want me to get started on the salad?

A: Can you pull together a list of all of our clients in that region for the past two years?

B: Of course. I'll get started on that right away!

Just use "start" to talk about ongoing actions:

I need to start exercising.