My question is, what is the newest(best, and easiest?) way to make a responsive web layout.
That’s not really a proper question…the answer is “all of them”.
There is no “one way” to create layouts because the layout (and your layout choices) will (or at least should) depend on the actual design.
Indeed a mixture of layout methods in a website is common…something needs to wrap around, use a float. Simple div centering, use inline-inline block, need a div to be independent of other divs in a specific spot on the page, absolute or fixed positioning.
Only supporting modern browsers and want full flexibility, perhaps flexbox is the way to go.
Show us the design, support requirements and you’ll get a variety of methods to achieve the same thing.