The proper way:
1) Put the bombilla into the gourd.
2) Put the mate in the gourd.
3) Invert, shake vigorously, revert. If possibe, make a small "mountain top" on the oposite side where the bombilla enters the mate tea.
4) Add a small amount of cold water (this isn't necessary--as I understand, this step helps to keep from burning the mate, which can also be accomplished by using water under 180℉~82℃), wait for a few minutes, until the mate absorbs the water.
5) Add hot water, drink.
6) Repeat step 5.
Don't move the bombilla while drinking, as this will cause you to stir up little bits. Inverting and shaking isn't necessary if the mate is of high quality (low dust). You can make mate to low dust by putting it in a metal sieve and shaking. Save the dust and small leaves that fall out--they can be used to brew mate in a coffee pot.
It is also worth noting that you can brew mate in a coffee pot or a french press and achieve similar results, although the resulting beverage isn't quite the same as drinking from the bombilla.
Also, the second most common way to prepare it is like plain tea: put some mate tea in a paper bag and steep with hot water for several minutes. This is called mate cocido (cooked mate) and is not uncommon to offer it to children.