Upgrade gear is not a very good term to use in Diablo. Gear upgrades could have multiple meanings. Are you trying to upgrade it by getting a set, just stat upgrades or a specific item? All of these would be valid for "upgrading gear".
I would first suggest pinpointing what kind of upgrades you want to get. You should first aim at a specific build that you want, and then looking for that.
As for your question, here we go:
Crafting recipes in Diablo are generally useless unless you have really weak gear in a specific slot. I'd stay away from them because you can use salvaged parts in the cube.
Bounties are good for cube materials, which will ultimately help your build, but not your standard gear items.
Blood shards are great, but stay away from weapons and jewelry. They are too expensive and are not cost efficient while first upgrading your gear for the late game.
There are no specific bosses that are better for gear and bosses are not worth it.
When you are looking for a SPECIFIC piece of gear, the fastest way to obtain it is to turn a rare (yellow) item into a legendary of that type through the cube. You can view the cube recipe's by clicking the little book icon in the cube menu.
The best place to upgrade gear is rifts because there is a bonus to legendary drop rates. You should always go to a higher difficulty if your clear speed will remain high. Always go down a level if your clear speed drops to something that is not time efficient.