I recommend a Background Summary instead of an Objective. An objective is about what you want...an background summary is about what you offer. Makes a difference.

Summarize you, your skills and abilites. For example..."energetic, hardworking,...blah blah...looking for an organization where I can increase my knowldege while helping the orgaization meet its objectives...."

An objective statement gives companies a reason to weed you out rather than in...if your objectives don't match.
Both a hospital and a store would want someone who is friendly, customer-focused, able to work with minimal supervision. Attention to details, etc.

Cover letter should always be included. You don't need to address why you are looking. What you want to say is why you find THIS opening interesting and why you should be a candidate they interview.