Advantage plans are typically HMO-type programs where you select a primary care physician and are referred to specialists and hospitals by that physician. Kaiser Senior Advantage is the primary example of this that most people know about.
A Supplement plan allows you to any provider nationwide as long as they accept Medicare patients. A Plan F is the most comprehensive Supplemental plan and covers 100% of what Medicare Part A and Part B do not cover for any claims that are a Medicare-permitted expense. There are other options to reduce costs.