When you register for classes, check the names of the professors. Any course that lists "staff" has not been filled yet and will almost definitely be taught by an adjunct faculty member.
If a professor's name is listed, check the faculty websites. Most universities and colleges now provide faculty with websites where you can find information from class syllabi to office hours. A few hints that a faculty member is an adjunct are lack of office hours or no listed office. Most full-time faculty will have an office on-campus.
Call the dean or department chair of the department in which the class is housed. Colleges will not advertise who is an adjunct and who is not but, when pressed, they may provide that information. You can also ask which professor is the best fit for your specific needs. A permanent instructor with substantial experience will normally be recommended.
Talk to your adviser. College advisers know who's who on the campus. While they may advise you that adjunct faculty are just as strong as other instructors, if you express your desire to be in a permanent professor's class, they can steer you in the right direction.