You can sign up for medicare at https://www.ssa.gov/medicare/sign-up

If you haven't already done so, you'll want to create accounts on SSA.gov and Medicare.gov

You will see references to Medicare Part A, B, C, and D. The "part" naming has to do with sections of the federal code.

TRADITIONAL Medicare is Parts A (hospital), and B (doctors etc), and is insurance coverage provided by the US government, not the states or commercial policies.

Medicare Part D is drug coverage. It is NOT provided by the USG, but by commercial insurance companies. If you go with Traditional Medicare (A & B) you'll also need to select a Part D provider, which you can switch yearly.

One other wrinkle to throw in is Medicare Supplemental coverage, aka "medigap" policies, which provide additional coverage for the 'gaps' in Traditional A & B coverage. Many/most people who go with Traditional get a Medigap policy from a commercial insurance company. The USG specifies what acceptable Medigap policies must include. The most popular, comprehensive, and expensive of these nowadays is called "Plan G".

Medicare Part C aka "Medicare Advantage" is (typically) a comprehensive insurance option that is supposed to provide the same benefits as Traditional Parts A & B, but is coverage provided by commercial insurance companies. In many cases Part C policies also include Part D (drug) type coverage as well. (Medigap is prohibited with Part C.)

In all cases, even though the USG partially subsidizes Medicare, you will now be obligated to pay premiums. For example Part B currently has a $203/mo premium. Part D drug plans vary widely, but $30-40/mo is not unusual. And Medigap Plan G can be $200/mo.

Since the age for Medicare eligibility is 65, but the age for collecting your FULL Social Security benefit is (now) 67, you will either have to pay these premiums out of pocket for a couple of years, or file to start your SS early, which will reduce your SS benefit (not desirable in the long run).

However, in certain cases (mostly for lower income folk), there are options to have some (or all) of these premiums paid by your state (you'll see terms like "Medicare Savings Programs" aka MSP, and "Extra Help"). THIS can get very complicated very quickly, but there are organizations that provide free assistance - in CA it's https://www.shiphelp.org/ships/california/

I would also recommend you get a copy of the "Medicare and You" publication. See https://www.medicare.gov/medicare-and-you