rifts are specials realms which you can open in town (next to oric in adventure mode), same for grifts (greater rifts). In order to open grifts you need to find grift keys which can drop at the rift guardians in rifts (the higher the torment level the more likely a key drop is).
rifts/grifts are the best way to farm items and exp effectively.
shards are blood shards and can be spent at kadala. most likely you ll get yellow or blue items but she can also give you legendaries or ancient items (so a good method to farm specific items but beware some rare legendaries or sets are RARE ;) ) The more a class needs a specific item the more likely it is that you don't find it :P
legs are legendary / ancient items
best way to find items is to farm bounties or rifts/grifts. the higher the difficulty the better the drops are.