A Checklist is a list of items to be verified at the end of a sprint, release, or some kinda of milestone. Checklists are typically very straightforward checks to ensure that your testing is complete and robust.
Ex. Think of a checklist like Agile's definition of 'Done-Done Criteria'
A test scenario, is a kinda of use case that includes a list of test steps which are each independently verifiability that relate to the specific scenario. A test scenario should, where possible, include a user persona.
Ex. As a Manager, I expect X to occur when Y is in place = Test Scenario; Step 1, 2, 3, etc are checkpoints that relate to verifying the functionality expectation.
@mega, being the first person in the company, I would identify the high-level Checklist items first as this can act as your foundation for implementing an overall quality strategy. Then I would start focusing on test scenarios but really you need both.