The word “doing” is a gerund! It’s a verb that ends with “-ing” and can now function as a noun. Both terms are correct so you can use whichever fits the context. We can “start to do” something OR talk about something we wanted to “start doing.” Hope this helps :)

Hi there "start to do" and "start doing" are one and the samething.

There is no difference. They have the same meaning and are interchangeable.