Who initiates more relationships during the series?
Mat. Rand and Perrin don't initiate relationships. They wait until prospective partners tell them they're dating.
Mat is the only one who initiates any relationships. Although it's worth noting that his relationships tend to be of the one-night stand variety until women start initiating relationships with him. Also, Mat's life has left his personality somewhat different from what it had been in the Two Rivers.
Rand does better (in quantity) than Perrin in that
Rand ends up with multiple wives in the Aiel tradition.
But Mat is the only promiscuous one.
Of course, Perrin and Rand may have more sex than Mat. Mat is constantly between partners until women start claiming him. While Perrin and Rand have fewer partners, they are never between partners, just separated from them.
None of them ever have any particular success communicating with women. Women communicate with them instead. In rare cases they understand. More commonly they find themselves asking some older male friend to explain things to them. Mat might be best at this, but of course there are cultural differences with the woman that he is fated to marry. His extra memories don't help him much with that.