How long did it take to earn and gain respect from supervisor so that he/she was confident that what you suggested was the right approach?

I guess it depends on the nature and personality of the supervisor and your discipline. Essentially, your PhD is your research apprenticeship so you are learning all the time from the supervisor. However there comes a point in your PhD where you are expected to carry the project forward yourself and it's you who is leading the supervisor. The time for this is probably discipline-specific. In the sciences, this may be later in the PhD. Don't be afraid to disagree with your supervisor so long as you can back your ideas up. It's good practice for the viva anyway. Sorry if this sounds somewhat vague but it's a student-, supervisor-, and discipline-specific thing.