The U.S. Medicare program is complex and people just starting out can find it overwhelming. The two options generally available (besides basic Medicare coverage) are Medicare Advantage and Medigap. Each state can make their own rules about plans, so what's available to you can depend on where you live.
Medicare Advantage is not Medicare, it is private insurance and comes with most of the baggage people may have experienced with employee sponsored health insurance. There are prior authorizations, limited provider networks, geographical limitations, significant fine print, HMO's, PPO's and insurance companies who can and sometimes do deny care. MA plans can offer perks like dental and optical coverage, gym memberships, and nurse and doctor hotlines. They generally include some kind of drug coverage.
Medigap works alongside standard Medicare and all coverage determinations are made by Medicare. The portion of a bill that is not covered by Medicare is automatically forwarded to the Medigap insurer and they pay it with almost zero exceptions. There are no provider networks or geographical limitations (as long as you're getting care in the U.S.). If a provider takes Medicare they will also take Medigap. A separate drug plan will probably be required. If Medicare doesn't provide coverage for a doctor or service, neither will your Medigap plan. Some of the plans also offer perks such as optical and dental coverage.
The single most important thing to understand when choosing whether to go with an Advantage or Medigap plan is the Medigap open enrollment period.
During the first 6 months after becoming eligible for Medicare you can sign up for a Medigap plan at the very best rates. There is no medical underwriting (health questionnaires) and you can't be refused coverage. In the vast majority of states that changes completely after 6 months and signing up for a Medigap plan can become difficult or impossible. In other words if you choose a Medicare Advantage plan you may not be able to switch to a Medigap plan after your open enrollment period ends.
Medicare Advantage costs less if you don't need care and in many areas of the country can be a good choice, especially if you're lucky enough to have world-class facilities and doctors as part of your Advantage network. If you need care MA can cost much more than Medigap plans and be more difficult to manage.
Medigap plans can have higher fixed costs but if you need significant care can cost much less than Advantage plans. For high coverage Medigap plans like Plan G once you pay your deductible ($233 this year), pretty much all care is covered.
Agents who sell Advantage and Medigap plans work on commissions and those commissions continue to be paid as long as you keep a plan. Commissions vary by plan so keep in mind that the recommendations you receive from an agent may be very much affected by their commission structure.
Medicare.gov is the best overall source of information about the program, but it isn't without its issues. The site can be difficult to navigate and it takes time to figure out.
States also run SHIP - State Health Insurance Assistance Programs. A directory can be found at shiphelp.org.
These are good places to start.
The complexities of Medicare and the various associated programs are hard to overstate so these few paragraphs leave a lot out, but they're also a start.