Say your Sim joins Entertainer, a base-game career with Musician and Comedian branches. Open the briefcase after accepting the job. If the current daily task says Practice Music, go to Build Mode, open Hobbies and Skills, buy an affordable instrument, then return to Live Mode and click that instrument to choose Practice. Check the briefcase again to see whether the task has registered.
If the task instead says Write Jokes, click a computer, open More Choices and choose Write Jokes. The exact daily task depends on the current level, so do not follow an example blindly: match the action to the words on your own Career Panel.
The same reading method works for other careers. Find the object or menu named by the task, perform the action, check the panel, then complete the listed skill requirement. When the branch prompt eventually appears, read both paths before choosing. There is no universal “always pick this branch” answer for every household.