Is the guy supposed to plan the first date?

Let him plan the date if he asks you out. If a guy has manned up and asked you out on a real-life, face-to-face date, then let him plan it. If he asks you from some suggestions, then by all means, throw some out there. By letting him plan the date, hes showing you hes manly, competent, and wanting to impress you.

Who should arrange the first date?

Traditionally, the guy would ask a girl out and then plan the first date. Lets take our parents as an example. Chances are they met, your dad asked your mom out, and he planned their first date all by himself - probably took her on a dinner or movie date.