Hello!
I'm sorry to say your question is very general - I mean there are many additional questions that need to be answered to give a better answer, like what airbrush do you use, what kind of paint and so on.
I'd like to give you some idea where to start, so first thing I'd recommend would be to practice painting thin lines - you could start with water based ink on paper to make cleaning the airbrush easier. When you get the idea about the lines you can draw this way, you could switch to the paint you would be using on the model - then it would be the time to experiment with the paint to thinner ratio. Now comes the big question: do you have a way to adjust how much paint goes through your airbrush?
If you have a "single action" airbrush with no way to restrict the paint quantity through it, painting thin lines is going to be very hard, if possible at all.
But if you have a "double action" airbrush, or a way to set a small quantity of paint to come out of the airbrush, you have to pay attention to the rule saying that the closer you get to the painted surface, the less paint you wat going through the airbrush. The paint also needs to be thinned more. Then you go really close to the surface and start spraying. You also want the airbrush to be moving constantly - when you stop the movement you will get a bigger dot of paint where the airbrush is pointing and it might start to run.
I would also say that unless you want very light or problematic colour (like yellow, orange or red) a light gray background would probably work better for you than white.
I hope it helps, if you have more questions please ask them - good luck with your builds and have a nice day
Paweł