See what an unsatisfying bumper-sticker treatment that was? Among the different roles assigned to men and women in the Bible, men are assigned the role of leadership. This is not a signal of male superiority or of the greater importance of men.
It is simply God's design and assignment of equally valuable roles among spiritually equal beings. Briefly, biblical support for this position is found, among other passages, in the creation order in Genesis 2, in 1 Corinthians 11: 7-9, and Ephesians 5.
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?
As you move into the stage of life in which you begin to seriously consider marriage generally or a particular relationship, your first step should be to soberly reflect, before God, on your own spiritual walk and maturity in Christ.
True, these passages refer to marriage, but it is wise and right to set patterns that will serve you well in marriage, especially if one accepts the premise that the purpose of dating is to find a marriage partner.
What does this actually look like in a budding relationship between two people?
In my view, if you can't happily picture yourself married within a year, you're not in a position to date.
Third, once you decide that you are ready to date, look to God's Word to decide the kind of person to date, and evaluate potential dating partners on those criteria, rather than relying primarily on the world's treatment of ideas like "attraction" and "chemistry." I wrote at some length on this in my article, "Brother, You're Like a Six." For you busy singles with time for only one mildly irritating column per day, the summary is this: Pick a potential dating partner with an eye toward godly manhood and womanhood — with an eye toward who would make a good husband or wife, defined by those characteristics esteems in His Word, not the ones Hollywood likes.
First, the man should initiate asking the woman out.