entry level health care field options?

Nursing is entry level.  I went to university so...competition was low when everyone is the same.  Nurses work 12 hour shifts on the ward.  Blood lab work 10 hours, everyone else works 8.

Of the options you listed, CNA is the most relevant. Skip phlebotomy. It's at best a technical motor skill you will use very little unless you're in a specific role as a nurse. Nurses who need to start IVs get good at starting them. 

EMT is great (and you'll learn more about assessment) but it is totally different environment and mindset from nursing. Even the ER is not the same as the back of an ambulance. It's great experience and it is certainly valuable but it is still not nursing. 

As a CNA you have the most direct exposure to what nursing really is and the different kinds of nursing. Nursing is not just a technical motor skill and you don't develop specialty expertise until well after graduating. Learning to be a nurse is about learning a particular way of thinking and approaching problems. CNAs work directly with nurses so you get much more opportunity to see what nursing really is, how different nurses practice, the different areas of nursing, etc. 

As far as getting into school goes, realistically either EMT or CNA experience will count. It's much more about having spent time putting your hands on people than it is specific tasks that you have experience with. Nursing school will teach you how to be a nurse. The experience you get beforehand should be to teach you about whether taking care of people is your path or not. More than one person has gotten into school, gone to clinicals, and only then found out that the interpersonal aspects of the job are a problem they can't overcome. 

A CNA.  EMT is a different skill set and the hours can be 24 on, so hard to do nursing, and it’s g

Hard to find jobs in some markets.  Phlebotomy doesn’t really expose you to nursing at all.  CNA at least you can talk to nurses, experience different floors to get a feel for what you might want to be a nurse on, and get nursing skills or at least observe many, as well as learn to prioritize and manage patients.