steps to do this, considering vario us siding materials such as vinyl siding and t-111. thanks

I would nail 4'x8' insulation board and then hardie board over it, good stuff that hardie board.

the problem with asbestos siding is that it breaks real easy. so to fasten another siding over top of it ,is pretty difficult, normally the asbestos siding is removed, and disposed of properly. you could try something that you could screw on , drilling pilot holes first , would be real time consuming, the other option would be using furring strips then fasten your siding to the strips.could even use some foam board insulation between the furring strips for added insulation.