What to do with extra rare/set/legendary items in Diablo 3?

Rares should be salvaged, always. They're completely useless once you fill every slot with a legendary item. You'll need the materials you get form scrapping rares (and also white/blues) in almost every recipe.

For the majority of the legendaries you find, their best use is to either extract their power using kanai's cube, or break it down into a forgotten soul. The one caveat is that it worth it to save any ancient gear you find just in case it comes in handy down the line. Ancient legendaries are rare enough that they're worth keeping.

For extra set pieces, there are ways you can use the kanai's cube to reroll it into another piece in the set. (10 souls, 10 dbs). This allows you to complete sets much faster.

Finally, there's almost no way to transfer useful gear between classes. Unless the item rolls near perfectly and you can reroll the main stat (say from str to int), or you can reroll it so it doesn't have a main stat, it won't be as useful as any legendary found on the wizard.

Even between same main stat classes, they typically use drastically different gear you won't be switching much other than a ring or two.