The idea was to protect the woman from potential hurt or awkwardness, to aid her in evaluating a man whom she might not have known well at the time of his initiation, and to help ensure that the relationship was carried out honorably.
Certainly, this norm spread beyond the believing community and became more of a cultural phenomenon, but it still gels well with attempts to carry out a godly dating relationship — especially among those believers who hold a complementarian view of biblical gender roles.
First, the man should initiate asking the woman out.
Whether this means approaching the woman herself or her father or someone filling that role instead of her father, it should be the guy that starts things off.
What does this actually look like in a budding relationship between two people?In this day and age, however, the hard fact is that many single Christian women have fathers who are not involved in their lives at all, are not believers, or are indifferent to or unaware of the notion of protecting and shepherding their daughters and potential suitors in a dating context.Where that is the case, a natural alternative might be some married individual or couple within the woman's (or man's) church community.If you don't desire that sort of protection or aid, at least insist that the two of you begin to meet with others who know one or both of you well so that there will be consistent accountability and an outside perspective on how the relationship is going.
Humble openness to accountability is essential to a godly relationship. It's not exhaustive coverage, I realize, but this should at least get your relationship started on broad principles. All singles who profess Christ and aspire to marriage — even as a possibility — have this responsibility (even outside this area of life, we should all be trying to grow in Christ). If you're already sure of that basic answer, are you a growing and mature Christian?