If you do get a wig cap, be sure to get some hair pins! Not bobby pins, but actual hair pins. If you use those and pincurl your hair under the wig cap you might be able to get your hair much flatter under the wig.
Also, a wig can be pinned down with hair pins! Pin the front of the wig and then pull the wig down tight over the rest of your hair and pin it everywhere else around the edge!