I'm still learning with my airbrush (had one as a kid, but that was a long time ago and I wasn't that good anyway).
However, I'm finding that it is much easier to get the hang of fine lines if you start playing with it on a porous surface, like paper or cardboard. Thin the paint way, way down. Like 60% thinner/40% paint or even thinner. Turn the air pressure way down to like 10 or 15psi.
Now open the nozzle a little bit, and get some paint flowing. (look at it from the side, against a dark surface, with a bright light behind you) If it won't flow, open a little more or add air pressure until it does. Just very little changes in each. Play with the variations of those adjustments. Play with distances to the work.
It should be easy to do thin lines with the paint that thin and a porous surface. Once you get the hang of that, you need to try doing a non-porous surface. You will likely need to thin less and add more pressure, but now you have some practice and a better idea of what you and the brush are capable of.
Use old soda bottles for practice (A trick I learned here). It helps if you prime them first, makes the paint stick better and easier to see.