How do I determine what their needs are and make sure they are being taken care of?

LISTEN well and ask questions.

Ask the patient and review the chart. Therapy, wound care, doctor, etc., should be documenting and writing orders. Insurance will determine providers the patient can choose from.

You can identify the patients needs while you are completing the initial case management assessment.
Also I always follow the therapy recommendations prior to discharge. This way if they need home care or DME it will be ordered & in place prior to discharge.

By being closer to the patient so that they can open up to you. Meaning you have to build good rapport so that the patient trusts you to tell you how they are feeling both physically and psychologically because a nurse is an advocate.

I’m assuming your referencing hospital care.
If so, their admitting diagnosis is the primary concern at hand. Identify what chronic conditions they live with - read your orders to
make sure you’ve got a grasp on the overall treatment plan. March ahead by knowing what tests/labs etc are indicated for specific conditions and you’ll see a lot of certain diagnoses so you’ll recall.
Find a good mentor and enjoy!