Dave Gage of the Harmonica Lessons website recommends that new players start with a "harp" that is a standard 10-hole diatonic harmonica in the key of "C". His recommended brand is the Hohner Special 20. It is easy to play, sounds great, and is reasonably priced around $20. The Special 20 can be used to play anything from Country to the Blues.
A diatonic harmonica is an instrument that typically has 10 holes and plays a scale in the key, or pitch, that it is tuned to. Harmonicas come in 12 keys, but the key of "C" is a great place to start. It's not too high, and it's not too low. It works with many popular tunes. Part of learning to play a harmonica is the pleasure of doing it. "C" will sound good and allow you to participate in more jams.
Although you're just starting out, avoid really cheap "harps." They often have leaky combs. The comb is the part of the harmonica that the air passes over to make sound. If it leaks, the instrument can be difficult to play, and it will sound bad. Go for a reputable brand, so you'll sound good from the beginning.
Always start with a new harmonica. Used "harps" are full of germs. They may also have damaged combs or be out of tune. It is hard to learn to play if your instrument is out of tune. You'll feel like you're playing the wrong notes.
As you become comfortable with the harmonica, you may want to try other brands such as Lee Oskar or Suzuki. Although they are more expensive, you may find you prefer their sound. Among the best harmonicas, there are slight tonal differences that appeal to a variety of musicians.