I know I go against normal convention in this department - but, here goes anyway!
I have done an extensive amount of work with inexpensive brushes, even though I also have some "good" brushes. I have a bunch of information documented in a FB page, including a write up on why I generally don't recommend one to a newbie. The page is DaveG's Airbrush Exploration (
https://www.facebook.com/davegs.airbrush.exploration/) Once on the page go to the tab called "photo's". From there, hit the "album" tab, and then "show all". If you click on each picture in each album, there is information tagged to almost all of them...
With the understanding up front, that purchasing an inexpensive brush can be a gamble ( I too usually recommend something along the lines of a Badger 105 or an Iwata Eclipse to beginners) these are some of inexpensive imports that I have chosen to keep, and use regularly - the most expensive was $46, most in the $21-$25 range
Creating textures with an airbrush is part of what makes them so fun to use! There are lots of ways of doing it, some of which has already been mentioned in this thread. Check out the tutorials associated with this forum, by looking all the way up at the top of the main page, and following the link - there is some interesting and fun stuff to be found there as well.
Here are a couple of quick examples of textures produced using just the airbrush - that background was done by forcing the brush to spit by pinching the hose, and then rocking the trigger and releasing a little air. Used just an air blast to create some direction to the texture when the paint was wet, and I also used a paper towel to remove some of the paint from the surface of the synthetic paper I worked on.
While the colors on this one may distract from how much texture there really is in this piece, it was all airbrush. Most of it was done by stippling, either by pinching the hose, or reducing the air pressure to get the result I was after. There was some stencil work also done in this, using FX texture effects from Artool -
Good luck on your quest! Will be looking forward to watching your progress.