I am worried that any answer to this question will be opinion based, but I will take as objective of a stab at it as I can.
Given that all human relationships are kinds of friendship according to Aristotle's definitions of the word, as some have pointed out in comments, we ought to consider the ends of friendship. According to Aristotle, there are three kinds of friendship: useful friendships, pleasurable friendships, and true friendships. True friendships are oriented towards the betterment of one another, growing in virtue, and they require that the friends love one another for their own sakes and are similar in virtue. Scripture here can also be cited, as Paul tells us we are not to be unequally yoked (2 Cor 6:14). In context, he is talking about being bound up with unbelievers, but this especially applies in marriage. Non-believers and believers are necessarily disparate in virtue. In fact, in looking for some Aristotle sources on this, I stumbled upon a great article that views marriage through Aristotle's understanding of friendship.
So, to the question at hand, courting is nothing if it is not an activity ordered towards finding a suitable partner in marriage. What precisely a suitable partner in marriage is depends upon what marriage is, ideally. And, if we understand that marriage is, ideally, a true friendship that goes beyond mere friendship and is also ordered towards procreation and child-rearing, then we can start to get a clear picture of what courting is in practice.
In practice, courting would be the discernment of whether or not the two suitors are, first of all, capable of a true friendship. So, they must be able to love one another for themselves, and not for merely pleasure or utility. They must be similar in virtue - one cannot be far more or far less virtuous than the other. And, they must have a desire to grow in virtue.
Those three points established, there may be other particular concerns of the individuals which must be discerned. For instance, they may want to be sure they agree on things like financial philosophies, parental techniques, etc.
The three required points are a general pre-requisite for any marriage, since any marriage which is not a true friendship is doomed from the start. Agreement on particular concerns are a pre-requisite for particular marriages. Courting entails discovering, both through conversation and mutual observation, whether a potential marriage partner meets the the required points as well as all of the particular concerns of the suitor.
Differently from modern dating, once both suitors have established that the other meets all of the general and particular requirements, there is no reason to delay marriage (generally speaking; it is possible that particular grave circumstances, such as no means to provide for a family, may require a marriage be delayed). Delaying marriage without a grave reason only invites temptation and occasion to sin.
I fear I may have made courting sound like a dry, mechanistic process, but it is not. When I said the suitors must discern whether their potential marriage partner meets all of the general and particular requirements for marriage through conversation and mutual observation, I was referring to mutual activities that facilitate this. Dinner dates, walks in the park, etc, etc. All the sorts of things that modern dating couples do, minus those that are sinful or constitute an occasion of sin, are permissible. And, those activities which facilitate the discernment of suitability are to be preferred.