Covering it up with the new siding is always the cheapest and safest way to do it.
If you do cover it up you can do all the work yourself.
If you remove it you may have to get a professional in to do the job. (depending on the regs in your area of course).
I used to have an asbestos removal license and I have put up a page on my site. This is Australian based, but I can't see US conditions being any less strict than they are here.