Medicare for Veteran's.

You should enroll in Part A since it's premium-free and will allow you to get coverage at a non-VA hospital. Part B is not considered creditable coverage, so the VA does recommend you enroll so you're not penalized in the future and again, can get coverage at a non-VA doctor. This article breaks it down.

The VA does encourage you to sign up for Medicare - Part A (premium free for most people), Part B can also be also be advantageous as well as Part D. (both will have premiums unless you have very low income) 

The VA healthcare system does not bill Medicare for anything.  But having your Medicare coverage can open other medical doors to you if need be -

Reasons:

• We don’t know if Congress will provide enough funding in future years for us to provide care for all Veterans who are signed up for VA health care. If you’re in one of the lower priority groups, you could lose your VA health care benefits in the future.
• Having Medicare means you’re covered if you need to go to a non-VA hospital or doctor—so you have more options to choose from.
• If you delay signing up for Medicare Part B (coverage for doctors and outpatient services) and then need to sign up later because you lose your VA health care benefits or need more choice in care options, you’ll pay a penalty. This penalty gets bigger each year you delay signing up—and you’ll pay it every year for the rest of your life.
• If you sign up for Medicare Part D (coverage for prescription drugs), you’ll be able to use it to get medicine from non-VA doctors and fill your prescriptions at your local pharmacy instead of through the VA mail-order service. There’s no penalty for delaying Medicare Part D.

You will need to choose which benefits to use each time you receive care.

To use VA benefits, you’ll need to get care at a VA medical center or other VA location. We’ll also cover your care if we pre-authorize you (meaning we give you permission ahead of time) to get services in a non-VA hospital or other care setting. Keep in mind that you may need to pay a VA copayment for non-service-connected care.

If you go to a non-VA (or VA authorized) care facility, Medicare may pay for other services you may need during your stay. Or, if we only authorize some services in a non-VA location, then Medicare may pay for other services you may need during your stay. Check your Medicare plan, so you know which care locations and services you’re covered for.

VA Health Care and Other Insurance 

@Graduate2010 some of this reply will overlap the reply by @GailL1 and some will enhance what she posted. Put the two together and you should have a good picture of the way Medicare and VA benefits work.

If you have or can get both Medicare and Veterans’ benefits, you can get treatment under either program. Generally, Medicare and VA can’t pay for the same service or items. Medicare pays for Medicare-covered services or items. Veterans’ Affairs pays for VA-authorized services or items.

When you get health care, you must choose which benefits to use each time you see a doctor or get health care.

Note
To get the U.S. Department of Veterans Affairs (VA) to pay for services, you must go to a VA facility or have the VA authorize services in a non-VA facility.

If the VA authorizes services in a non-VA hospital, but didn’t authorize all of the services you get during your hospital stay, then Medicare may pay for the Medicare-covered services the VA didn’t authorize.

For active-duty military enrolled in Medicare, TRICARE pays for Medicare-covered services or items, and Medicare pays second

For inactive-duty military, Medicare pays first for Medicare-covered services and TRICARE may pay second.

TRICARE pays first for services or items from a military hospital or any other federal provider

https://www.medicare.gov/supplements-other-insurance/how-medicare-works-with-other-insurance

In the FWIW department, I helped a Vet secure coverage earlier today. He is turning 65, has A & B and has used the VA for a dozen+ years for primary and specialty care as well as Rx.

Due to COVID most of his VA "appointments" are virtual and he wants the ability to see providers in person, in the private sector in lieu of using the VA teleconference system. He has a number of health issues, compounded even more by a recent stroke. Virtual appointments are not his friend.

He selected a Medicare supplement plan which, in conjunction with original Medicare, allow him unfettered access to health care. No networks, no referrals, no pre-authorization.

The supplement plan limits his OOP to the Part B deductible.

He also opted for a low premium ($7) PDP so he can use a retail pharmacy when needed. Most of his med's (including insulin) are SHIPPED to him by the VA clinic. VA filled Rx has very low copay's, especially for insulin.

He isn't my only VA/Medicare client, just the most recent one, and his situation is not unique.

One more thing . . .

Medicare A or B IS creditable coverage however the following are NOT creditable if you are applying for Part B outside of your IEP.

Examples of Coverage that is NOT Creditable Under Medicare
TRICARE
Veterans Benefits
COBRA
CHAMPVA
Retiree plans
FEHB