Me and a bud finally played a full game a while ago. In it we both thought that we could only use core troops (ex. storm, snow, and shoretroopers) to obtain objectives. Is this wrong? Can say death troopers or scouts obtain objectives? Is the "trooper" the only important part?

Objectives generally don't care what your rank is. Some of them specify "trooper" only, which does include units like Death Troopers, Scouts, and even Darth Vader and Luke. As long as the word "trooper" appears in their unit type (underneath their image on the unit card), they can score it. Some objectives don't specify a unit type at all, in which case all units are eligible, including vehicles.

Unit type and unit rank are completely separate qualities.

The unit type is different from the unit rank, so basically if the objective states trooper unit then any Commander, Operative or Soldier type unit can gain claim .*

The unit card itself specifies what type of unit it is, trooper, vehicle, creature etc. So Deathtroopers, Scouts etc. all count.

*Just be aware that there are some sub-types of units, such as emplacement troopers. They operate as trooper units with some extra limitations. One of those limitations is :" Emplacement troopers cannot use the claim or sabotage/repair abilities" (Rules Reference Guide, pp 39-40)