There are no current internship openings.
Generally, career librarians obtain a Master's Degree from a library education program accredited by the American Library Association. Although these programs have different names at different schools, such programs are generically referred to as Masters in Library Science degrees, or MLS for short.
There are two area universities offering Master's degrees in this field:
You might also wish visit the American Library Association's  career resources page.
Please note that while Kansas City Public Library offers many positions not requiring advanced education, positions in the Librarian/Professional classification do require the MLS or its equivalent unless stated otherwise.