Stasik P. answered 10/12/20
5th Year PhD student in Organic Chemistry at UCLA
Tertiary alcohols can't be oxidized to carbonyl compounds in general circumstances, since that would yield a carbon with 5 bonds.
As for primary and secondary alcohols, some common reagents for oxidation are pyridinium chlorochromate (PCC), Oxalyl chloride+dimethylsulfoxide+triethylamine (Swern), Dess-Martin Periodinane (DMP).
Hope that helps!