The question-title of this post is often asked to me periodically. I am generally asked if I would be open to dating/marrying a girl who wishes to convert to Catholicism.
I am usually reserved on this question because priests are not supposed to book a wedding until he is proven that both parties are Roman Catholic and are free to marry in the Church. (My priest will not book a wedding until such questions are answered.) But if there is someone who wants to date me, they will either (1) have to be Catholic or (2) willing to convert.
If a young woman converted before she meets me, then she is all set. But if she is willing to convert to Catholicism to see me and for the salvation of her soul, then it will depend where she is converting from. If she was baptized "In the Name of the Father, and of the Son, and of the Holy Ghost" with water outside of the Church, then this baptism will be recognized as valid with proof thereof. But if she is converting from a non-Christian religion, then she will definitely have to be catechized sufficiently in order to be baptized as an adult.
In the diocese I currently live in it is almost impossible to find a traditional priest who will administer the sacraments the right way, never mind proper catechists. If need be I am willing to catechize a future spouse if warranted. (I will make sure she will not go through an RCIA program!) There is no church law that says who exactly can catechize, albeit the teaching responsibilities of bishops today are so low that I had to learn the Catholic Faith (as a cradle Catholic) all in my spare time, thanks to the Internet.
As far as adult baptism, this can be done by any priest at any time during the year (not just at the Easter Vigil). But if a priest cannot be reached, then any layperson can baptize. It is quite possible that a layperson will have to perform the simple baptism as noted above, but I nor my family members canonically could not do it because doing so would constitute an impediment toward marriage for me because of the spiritual relationship between my relatives and her.
As far as Confirmation is concerned, I will bend backwards to make sure she receives the Sacrament in the Tridentine form and by a bishop. (There are no resident bishops who will do this in New England.) She will have to find a female sponsor for Baptism and Confirmation.
Lastly, no wedding plans can be made until she receives Baptism. (Confirmation may be delayed for a good reason.) I hope she would not convert to try to ease the rules. For some inspiration, here is the Wikipedia list of some famous Catholic converts.