When my city gets large enough to get crematoriums and incinerators, I generally use those over cemeteries and landfills. They don't need to be emptied, and I'm going to need the burners anyway to empty the other buildings. Invariably, my city winds up with dead bodies (and trash) in almost every building, so that got me wondering if I do need cemeteries to perform the initial body pickup? And crematoriums are only used when emptying cemeteries? Or do crematoriums actually collect dead bodies from buildings, and my large number of uncollected bodies is indicative of another problem (traffic?)?
Yes, you need both cemeteries and crematoriums, as some citizens will refuse to be cremated. (Source : Cities Skylines guide)
For trash this is not necessary, you can suffice with just incinerators as trash doesn't really have an opinion on how it is disposed off.
For both it is important to build enough facilities spread out trough the city as they have a limited number of trucks/hearses and these need to drive trough traffic to the pick up location.