Why can't I see my outcome in the list of outcomes to link to an objective?
Check that the plan that has the outcome(s) in has been published, else linking won't be possible.
What sort of thing should I put into the objectives?
Objectives are designed to sit above project specific outcomes and be more focused towards business or portfolio goals. You can link specific project outcomes to the Objectives across all projects which allows you to track the progress to meet that specific Objective.
Within an Objective you add key measurable results and assign them a confidence level along with an owner. The Objectives will show on the Project Overview page to keep the project team focused on what they're trying to achieve.
Some examples of objectives are:
Operational efficiency increased by 50%
Capture a bigger market share
Consolidate global business units into sectors