I agree that your resume should list the care you provided your family members as though it were working for a staffing company. It is all in the way that you write it!
For example, “Private duty Home Care RN- 2010 to 2012. Provided home care to two patients with multiple chronic diseases. Assessment, care planning, supervision of non-licensed caregivers, medication administration, skin care, wound care, care coordination, communication with health care providers. etc, etc”
If you are serious about wanting to work in home care, don’t let a poorly organized orientation be the reason you leave a job. Stick with a job, learn all you can, and perhaps after you have been there a while, you can make improvements to the orientation process. If your orientation was not adequate, then politely ask the employer for more training.
There are a lot of resources on the web for learning, so if your employer does not provide what you need, educate yourself.