Yes, but you can speed up the process a lot by learning from others. There are many great harmonica tutorials on youtube, for example. Different teachers work on different styles of playing and different genres of music, but the most common is blues harmonica.

With regular deliberate practice, you can expect to be playing simple pop tunes within around 3 months. Within 6 to 12 months, your technique will improve and you will probably be able to work on bending notes (a very important skill for getting the best out of a harmonica).

The Diatonic harmonica standard 10-hole is the best for nailing that wailing Blues sound, and it's the most common harmonica heard in American popular music. It's also the easiest harmonica on which beginner's can learn to play and start sounding great right away.