IMO it depends on your view of a relationship. For some, a relationship is a precursor to marriage and for some it's just dating.
I'm one of those who believes that it's a precursor to marriage so I don't wait until the relationship begins to bring it up, so it's brought up quite early. I don't ask about marriage directly but more holistically like where do you see your self x years from now. What are your views on marriage, sex and children etc.
Yes I'm pretty bold but if you both are on the same page it doesn't seem as intimidating.
I think that it's not necessarily about who goes first but if you want to know you should ask. IMO, you need to know if you're wasting your time upfront.
I don't think that you should nag your SO about it, it should be something you both want as both see it as a natural progression. From my few relationships I've notices that if a man wants to marry you and he knows that you see it in your future, he talks about it as much as you do.