The shingles contain less than 1% asbestos fibers. It's a non-friable material UNLIKE the friable 'stuff' that they wrap old heat pipes with.
As long as you don't go smashing them up in an enclosed 'space' and breathing in the dust, you should be fine. Of course, wear gloves, long pants, and long sleeves. If you're replacing a few shingles, I see no need for a mask (again, assuming you're not busting them up into a million pieces w/a hammer.) Most municipalities allow residential homeowners 'dispose' of them in their 'regular trash'. HOWEVER, if you're re-doing the ENTIRE house, you should rent a hazardous waste dumpster.
As I mentioned, if you want to go the replacement-siding-route, e-mail me directly and I can advise. My best to ya and hope this helps.