Really, this question can also be answered from other similar viewpoints to Wesleyan, such as Methodist or perhaps Lutheran. The closer to Wesleyan, the better.
Anyway, just as the title asks: what is marriage? Is it the legal procedure and contract that binds two people together with the force of law? Does there have to be a ceremony (a wedding)? Do common law marriages count, where the couple acts as though they were legally married? How about other cultural wedding ceremonies, such as hand-fastening?
I did some Googling for "wesleyan common law marriage", and basically the only relevant result I found was this article (from English 205 at a Wesleyan college), which pretty much says that around the time of Shakespeare, common law marriages were looked down on by church people, possibly even considered invalid. 'Course, that was in the late 1500s and the Wesleyan Church started in the mid 1800s, about 250 years later. So I would expect to see that the Wesleyan attitude is different today.
To clarify the ideas I'm attempting to express here, suppose that you have two people, Adam and Belle, that are in common law marriages with Danielle and Charles, respectively. So Adam is "married" to Danielle and Belle is "married" to Charles. Now, if these common law marriages are not valid, then if Adam and Belle have sex, they are not committing adultery. Depending on how you define "extramarital sex", they would then be either having extramarital sex or premarital sex, and the morality of the latter is uncertain.
Aside from the legalities, what about the spiritual side of it? Suppose that two Christian strangers meet spontaneously and get married within the hour, with the legal procedure completely satisfied. (Maybe an hour is pushing it, but in as short of a time as possible.) Suppose that neither are under the influence of drugs or alcohol, so the consent given is valid. Ignoring how this would probably very much not be what God wants, is this marriage as equally valid as a "normal" one? Or does/should a marriage include both a legal process and a spiritual covenant? Or...even...is it possible to be married by spiritual covenant alone?
I ought to note that I am not interested in the issue of homosexuality. This question asks about what defines a marriage between a man and a woman.