There are a lot of different slang terms used for harmonica: harp, blues harp, tin sandwich, Mississippi saxophone, harpoon, ten-holer, mouth harp. Whatever you want to call it, learning to play involves getting pretty intimate with it. Like a new prosthetic limb, it takes some time for the brain to find ways to control the new extension. In this case, we need to control the breath, using the feedback from the sound the harp is making.
At first, your sounds are likely to be jerky, uneven, and not as pleasant as you would like them to be. It can seem like the harp just isn't functioning right, but it is more likely your brain and body have just not adjusted yet to this new device. After reflecting on my own progress, and hearing more newcomers, I think there are some ways to speed up this process.
Good tone is influenced by several elements but the top two are your connection to the harp (also known as embouchure) and breath control. We breathe and vocalize so easily by the time we are adults we take mouth control and breath control for granted. You will develop natural control of these over time, but to begin with, you have to do things consciously, making small changes so your brain can connect these with the sound that is produced.
Paradoxically, having a good instrument may require developing a bit more control to start with. A tight harp is much more responsive to breath pressure, meaning it takes very little air force to make sounds. With a leaky harp, you can breathe almost normally right through the instrument without getting winded or feeling breath starved. With a tight harp it can feel a little like breathing through a straw. Here is my advice for getting started on the right foot. As we say in Tai Chi - The Breath is the Handle!
Hold your harp so the low notes are on the left side.
Stick the left side of the harp into your mouth - don't try to daintily pucker like you kissing it! You really want to be blowing and drawing (sucking) through the 1,2 and 3 holes at the same time. On your C harp drawing on the low end will produce a G chord and blowing will produce a C chord.
Try just breathing through your nose at first, not making any sounds with the harp. Try doing this to a 4 count, that is as you breathe in count mentally "1, 2, 3, 4" and the same when you breathe out through the nose.
When you breathe through your nose only, the back of the tongue is planted on the roof of the mouth. Try to slowly let some air in and out of the mouth while still breathing through the nose. Your harp should begin to make noise now! This will take a bit of time to develop, so relax, it will come.
This allows some air to move through the nose while most is still going through the mouth. If you feel like you are getting too full ( or your sinuses are too stuffy to breathe through the nose) you can let some extra air out by lifting the top of your lips from the harp just a bit - this will allow you to let some air out while still making noise.
Practice this exercise for about 5 minutes and work up to 10 minutes - it will feel a little like meditation (and indeed it is a form). It takes time and repetition to get your breathing and the sound to get synched up in your brain. If you feel lightheaded, just stop and breathe normally through the nose until the feeling goes away and then start again.
As you cultivate this, try to get the same amount of time for the in and the out-breath, and the same volume from the harp. Next, try lengthening your count, so you are mentally counting at the same speed up to 6 on the inhale and the exhale. See how relaxing the chest and shoulder muscles helps here. If you can engage your belly and pelvic floor muscles this will also allow you to take much fuller, longer and deeper breaths. See how long you can get your count.
You can also start experimenting with moving your tongue forward and back in the mouth, and also dropping and relaxing the jaw to form a much larger empty space in the mouth. Note the different tones you're are creating! You can also drop the tongue almost completely so the breath is more connected to the belly without having to regulate the pressure with the throat.
Developing a relaxed slow deep breathing pattern will help you cultivate your tone and clearly play single notes later on. It will also help you sleep better when done regularly!