How do I find out to which benefits I am entitled?

SFC Terry Stinnett is absolutely correct. Start with a VSO or VAO as they will be the most knowledgeable about what benefits you are eligible for. For your situation (what am I eligible for), I would go with a State's Veterans Affairs Office* as they will be the most knowledgeable about federal and state benefits.

Regarding getting a Veteran ID card, it's all online now*. If you're talking about a military ID, then that would be at the nearest RAPIDS site*.

Your medical question is too open-ended. Are you referring to accessing Community Care where the VA sends you to someone outside the system? Are you referring to using other insurance? Etc.
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* VAO List - https://department.va.gov/about/state-departments-of-veterans-affairs-office-locations/
* Veteran ID from VA - https://www.va.gov/records/get-veteran-id-cards/vic/
* RAPIDS locator - https://idco.dmdc.osd.mil/idco/locator

I'd recommend you start with a VSO, Veterans Service Organization, in your community. Perhaps there is a liaison at your county government as well?