I am very sure that you are aware of what you are dealing with, right???
Asbestos, is an inhalent so always wear a mask when working around it. I am sure you are thinking, "I have lived here for soo many years and nothing has happened yet, so why worry". Well, before I am sure you have never been working so close and long as you will be now.
First, you want to power wash the siding and get the old residue off.
Second, you want to apply several coats of a good primer like Zinzer B-I-N. I would think that the surface will take alot of paint.
Third, I would get a good brand like Sherwin Williams, Super Paint, Exterior, Latex. Mix some Flotro in the paint by recommandations and this will make it go on smoother and fight chalking.
This time of year, make sure you do not paint in the full sun light. Paint the west side in the morning and the east side in the afternoon. It will dry or seem to dry fast in this temperature, but do not be fooled, it takes 24 hours to cure. So, don't apply that second coat right away.
DO NOT USE ANY PAINT FROM THOSE HOME CENTERS, YOU WILL BE WASTING YOUR TIME AND MONEY.
Others may add on to this asbestos thing. So be prepared.