The Crafting system in Diablo 3 is not very sophisticated compared to most other games. Yet, there are still some noteworthy items that make a difference in many builds, and which can generally be grouped into three categories:
supplementary sets (worn alongside other 6-piece sets)
niche Legendary items for specific builds
Most of these require a combination of some Crafting materials found either form salvaging items or a Large Horadric Caches found from Bounties. The different materials are: Khanduran Rune, Caldeum Nightshade, Arreat War Tapestry, Corrupted Angel Flesh and Westmarch Holy Water, two of which are typically used for an item (so in cases where you can choose which set item to craft, you could consider one that requires materials you have too many of anyway). In addition, they usually require a moderate amount of 15-50 Reusable Parts, Arcane Dust or Veiled Crystals.
Compared to trying to find items from Rifts or acquiring them from upgrading or reforging in the Kanai's Cube, it's very easy to get well-rolled ancient versions of crafted Legendaries due to how cheap they are in comparison. Because of that, you should always try to craft until you get a good one.
Before you can craft any Legendary or Set items, you must find a corresponding Plan and feed it to your Blacksmith. After your Blacksmith learns a Plan it can no longer be obtained, meaning that eventually you will find all the plans in the game and then they will stop dropping completely. Plans can only drop from Odious Collectors and Large Horadric Caches!
Craftable Legendaries and Sets
There are some more than the listed items, but most of them are very useless and don't have any special effects. In the following, see a list of all the relevant Crafting options in the game:
This set is generally useful for builds that lack a good Bracer option. By using the Bracers + any other set piece, combined with Ring of Royal Grandeur, it can fit in without breaking the typical 6-piece set bonuses that carry most of your character's strength.
Niche crafting option for Demon Hunters
In earlier seasons, this has been a core item for DHs using the Marauder's Embodiment (M6) set with Cluster Arrow, as there was no build-specific weapon for it yet (such as Manticore). These days, it can also be used as a weapon option for the Gears of Dreadlands (GoD) set, as this one also does not require a specific weapon to be used.
This set is generally useful for support builds that require every piece of Cooldown Reduction they can get. In many cases, it is used for standard support Monk builds using the Weapon and Shoulder slots (along with Ring of Royal Grandeur, as it fits in very well there.
This set is very useful for players that need a lot of Greater Rift Keystone, as you get 25% extra keys while wearing it. Each key that drops from a Rift Guardian inside Nephalem Rifts has its own 25% chance to add another one, potentially giving you up to 8 keys at once you can get up to 4 keys at once in Torment 16. Wearing the set every time will improve your average key count per Rift from 3.5 to 4.375, which allows keyfarming speed upwards of 100 keys per hour at a very efficient speed.
Unfortunately, wearing the set will break most other 6-piece sets, as there are no item slots outside of the typical Helm/Shoulder/Chest/Gloves/Pants/Boots pieces for this set. This means that for many builds, it will be impossible to use in solo farming unless they are very strong on single-target damage regardless of their 6-piece bonus. Some examples are the Shadow's Mantle set using Impale and the Horde of the Ninety Savages using Frenzy. Other builds that are commonly used with Cain's are Natalya's Vengance (combined with Gears of Dreadlands) using Rain of Vengeance and Hungering Arrow on Demon Hunter, or Monkey King's Garb (Sunwuko) using Wave of Light on Monk.
When playing in groups, you won't have to worry much about being able to finish the Rift Guardian. You can simply swap to the set from your inventory when it spawns and another party member is present to kill the boss for you. Like this, up to 3 people can comfortable leech on the Rift Guardian even on builds that would otherwise not work well with the set. For this to work, you have to be around within 2 screens distance of the dying Rift Guardian, else the effect won't activate.
Captain Crimson's Trimmings
Supplementary set for Cooldown/Resource-heavy builds
Captain Crimson's Trimmingsis one of the core crafted sets in the game, used in many different builds. Paired with Ring of Royal Grandeur, it can be combined with other full 6-piece sets by wearing the Belt and either Pants or Boots. It has very strong effective, buffing Cooldown Reduction (CDR), Resource Cost Reduction (RCR) as well as damage and toughness of the character. The damage bonus effect is typically not strong enough to make CDR a very effective DPS-roll affix on your items (e.g. instead of Attack Speed, Average Damage or Area Damage), but it does add an extra benefit when you would otherwise add a certain amount of CDR to your build anyway. The damage reduction effect from RCR is very useful, however, as with the set one RCR roll usually corresponds quite closely to the value of other affixes (e.g. All Res, Armor, Vitality) on your items, on top of adding the resource effect.
Niche Wizard item
If you want to go really fast on a Wizard, the Teleport Wormhole will is almost indispensable. If you can fit in this Offhand, you can use another rune to give you additional benefits. While it's usually quite difficult to do so, it is especially useful on Archon builds as it has the unique ability to improve your Teleport skill with the Wormhole rune, giving you even more mobility!
Core Shield Bash Crusader item
While the build itself is rather uncommon these days, Crusaders going for a build based on Shield Bash using the Roland's Determination set typically want to wear this crafted Shield.
Core Rat Necro item
This is a very strong item for resource recovery in farming builds. It's usually paired with a support class that can consistently spawn globes (especially zNec through , sometimes zBarb through Ground Stomp or Threatening Shout ). While it has been nerfed to have a 2 second Internal Cooldown, it's still a very solid option for Rat Necros in groups and potentially other really resource-heavy builds.
Supplementary Torment 16 set for faster Death's Breath farm
This set is very useful to acquire Death's Breath very quickly and can be fit into almost any solid Torment 16 builds with ease by wearing the Belt and one other piece along with Ring of Royal Grandeur. The set will straight up double your DB income, giving you either 6 or 8 drops per elite pack in T16. Like this, numbers north of 2000 Death's Breath per hour are quite achievable if you are farming at a very efficient pace, which would give you enough materials for more than 40 rare upgrades in the Kanai's Cube.
When in groups, make sure to stay close to your party members, as the effect will only activate if you are within around 2 screens of the dying elite.
Core Garg WD item
For Witch Doctors going for a Gargantuan build, this is going to be the Offhand you want. The cleaving ability is what's going to carry your AoE damage output, so this is a must-have, and can be crafted with good rolls quite easily here.
Other Craftable Items
There are also certain other items that are not crafted using Bounty materials at the Blacksmith or are in other ways not really regular items.
Outdated/niche amulet choice
These amulets have been very useful in the past, until newer and better options (Squirt's Necklace, The Flavor of Time, The Traveler's Pledge) had been introduced. Right now, there are no builds that would seriously use one, unless you are getting extremely unlucky with your other amulets and you somehow accidentally manage to craft a very good Hellfire Amulet. Supports in XP speedruns can make use of this item, because The Flavor of Time does more harm than good there due to knockback from the Speed Pylon. Through this item, it's possible to acquire a fifth passive for your class.
Just like with the Hellfire Ring, this one can be crafted using the Uber Materials gained from the encounters started by Infernal Machine of Regret, Infernal Machine of Terror, Infernal Machine of Putridness or Infernal Machine of Fright. These can only be activated in the Heretic's Abode, which is a little room in New Tristram, hidden behind a breakable door near the Mystic.
Niche support Barb choice for XP speedruns
In order to optimize your Experience gains in group speed Greater Rifts, support Barbarians (or potentially other support classes) can wear this ring for a tiny bit of extra XP. The bonus is very small (after penalties around 1% bonus XP from kills, or roughly 0.5% bonus XP overall), so it should never be your primary concern to wear this ring. It's just the last piece of the puzzle in optimizing your character.
Just like with the Hellfire Amulet, this one can be crafted using the Uber Materials gained from the encounters started by Infernal Machine of Regret, Infernal Machine of Terror, Infernal Machine of Putridness or Infernal Machine of Fright. These can only be activated in the Heretic's Abode, which is a little room in New Tristram, hidden behind a breakable door near the Mystic.
Staff of Herding
Grants access to the Rainbow level (Whimsyshire), which cannot drop the Rainbow Wings, but is commonly used to acquire the Rainbow Portrait
Plan: Staff of Herding
Ever since the introduction of the Rainbow Goblin, there has not really been a use for this Staff aside from some fluff. In earlier times during Vanilla Diablo 3, this was one of the top endgame farming zones as it had high quantities of easy-to-defeat, high level monsters. You need to gather the recipe from Izual in Act IV and then acquire the different parts throughout Sanctuary:
Wirt's Bell (Act II, sold by Squirt the Peddler)
Black Mushroom (Act I Cathedral Level 1, 2 or 4)
Liquid Rainbow (Act II, Mysterious Cave Level 1 in Dhalgur Oasis)
Gibbering Gemstone (Act III, Chilltara in Ice Caves Level 2)
Leoric's Shinbone (Act I, Fireplace in Leoric's Manor)
May the RNGjesus be with you in your crafting session!